Symbolic Computation of Exact Solutions

of Nonlinear Evolution and Wave Equations

by

Wuning Zhuang

ii

A thesis submitted to the Faculty and the Board of Trustees of the Colorado School of Minesin partial fulfillment of the requirements for the degree of Master of Science (Mathematics).

Golden, ColoradoDate

Signed:Wuning Zhuang

Approved:Dr. Willy A. HeremanThesis Advisor

Golden, ColoradoDate

Dr. Ardel BoesProfessor and Head,Department of Mathematical andComputer Sciences

æ

ABSTRACT

Hirota’s bilinear method for finding exact soliton solutions of nonlinear evolution and waveequations is discussed and illustrated. The MACSYMA programs HIROTA SINGLE.MAX andHIROTA SYSTEM.MAX are included. These programs automatically carry out the lengthyalgebraic computations for the symbolic calculation of one, two and three soliton solutions. Theprograms also allow to test if four soliton solutions exist for both single bilinear equations andsystems of coupled bilinear equations. The MACSYMA programs are tested by constructingexact solutions of various nonlinear partial differential equations from soliton theory, such asthe Korteweg-de Vries, the modified Korteweg-de Vries, the Kadomtsev-Petviashvili, and theBoussinesq equations.

Abstract iv

æ

CONTENTS

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2. Hirota’s Bilinear Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1 The f -Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 The Bilinear Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 The Bilinear Form of a Single Equation . . . . . . . . . . . . . . . . . . . . . . . 42.4 The Bilinear Form of a Coupled System . . . . . . . . . . . . . . . . . . . . . . 52.5 Hirota’s Direct Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3. The Hirota Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.1 Hirota’s Conditions and Soliton Solution for a Single Equation . . . . . . . . . . 103.2 Hirota’s Conditions and Soliton Solutions for Systems of Equations . . . . . . . 14

4. MACSYMA Programs for the Hirota Method . . . . . . . . . . . . . . . . . . . . . . 164.1 Special Features Used in the Programs . . . . . . . . . . . . . . . . . . . . . . . 164.2 The Algorithm of HIROTA SINGLE.MAX . . . . . . . . . . . . . . . . . . . . . 174.3 The Batch Program for HIROTA SINGLE.MAX . . . . . . . . . . . . . . . . . . 194.4 The Algorithm of HIROTA SYSTEM.MAX . . . . . . . . . . . . . . . . . . . . 194.5 The Batch Program for HIROTA SYSTEM.MAX . . . . . . . . . . . . . . . . . 21

5. Examples and Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.1 Test Results for Single Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 225.2 Test Results for Coupled Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 26

6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Appendix 34

A. Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35A.1 The Proof of Hirota’s N-Soliton Condition . . . . . . . . . . . . . . . . . . . . . 35

Contents vi

A.2 The Proof of the New Expressions for Hirota’s Operators . . . . . . . . . . . . . 37

B. Flowchart of HIROTA SINGLE.MAX . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

C. Table with Logical Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

D. Program Codes and Test Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43D.1 HIROTA SINGLE.MAX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43D.2 HIROTA SYSTEM.MAX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48D.3 Output of Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

D.3.1 Korteweg-de-Vries (KdV) Equation . . . . . . . . . . . . . . . . . . . . . 53D.3.2 Kadomtsev-Petviashvili (KP) Equation . . . . . . . . . . . . . . . . . . . 55D.3.3 Modified Korteweg-de-Vries (mKdV) Equation . . . . . . . . . . . . . . . 57D.3.4 Higher Order Ito Coupled Bilinear Equations . . . . . . . . . . . . . . . 60D.3.5 Bilinear Equations with a Parameter . . . . . . . . . . . . . . . . . . . . 62

Contents vii

I am especially grateful to my advisor, Dr. Willy Hereman, for suggesting this thesistopic and for providing invaluable academic advice. I am deeply thankful to have been thebeneficiary of his knowledge and guidance. I would also like to thank the other members ofmy thesis committee, Dr. Steven Pruess and Dr. Jack Cohen, for their careful reading andmost appreciated suggestions. Many thanks go to the faculty members of the Department ofMathematical and Computer Sciences for their excellent courses which have been and will everbe a solid foundation for my current and further research.

I greatly appreciate the generous financial aid from the Department of Mathematical andComputer Sciences at the Colorado School of Mines, which has supported my graduate studiesover the past two years. Special thanks goes to Dr. Pruess for additional financial supportwhile completing this thesis.

I also gratefully acknowledge my aunt Ms. Elaine Lai for her kind help.I dedicate this dissertation to my lovely wife Lixiu, a constant source of support and en-

couragement.æ

1. INTRODUCTION

‘Solitons’ made their first appearance in the world of science with the beautiful report [1, 2]on waves, presented by J. Scott Russell in 1844. While riding his horse along the Edinburgh-Glasgow canal in 1834 Russell observed “the great wave of translation” produced by a canalbarge. When the vessel suddenly stopped, a rounded, smooth and well-defined heap of waterrolled forward with great velocity. It continued its course along the channel for a couple ofmiles without change in form or diminution of speed. The solitary wave, so-called because itoften occurs as a single localized entity, was born.

Despite some attempts by Scott Russell to guess at the analytical form for the peculiarwave profile, his observation remained unexplained until 1895 when two Dutchmen, Kortewegand de Vries [3], derived their now famous KdV equation for the propagation of one-directionalsurface waves in shallow water.

The great discovery of the universal nature of the KdV equation and its remarkable proper-ties was to await another 60 years. In 1955, at Los Alamos, Fermi, Pasta and Ulam [4] carriedout some numerical studies in an attempt to understand the energy distribution over variouswave modes in a nonlinear one-dimensional lattice, consisting of nonlinear springs and identicalmasses. In contrast to what they expected, the nonlinear interactions between the modes didnot evenly distribute the energy throughout all the modes. The unexpected nature of theirresults stimulated more work on nonlinear systems and also called for a mathematical expla-nation. In 1965, Norman Zabusky and Martin Kruskal discovered the ‘soliton’ property fromthe results of the numerical simulations for the particle behavior observed in the Fermi-Pasta-Ulam experiment. Remarkably enough the mathematical model for this apparently unrelatedproblem led again to the KdV equation. Focusing on the possible interaction of two solitarywave solutions, Zabusky and Kruskal noticed a remarkable phenomenon: Whereas during theinteraction two solitary pulses behaved in a most nonlinear way, they emerged from their colli-sion with their former heights, widths and velocities. The only evidence of the interaction wasa phase shift whereby the larger solitary wave appeared to be ahead of the position it wouldhave been had it travelled alone with the smaller one behind. Solitary waves with this strangeparticle-like behavior deserved a special name. The name soliton was chosen because of theending ‘on’ which is Greek for ‘particle’.

The recent history of the mathematics of solitons begins in 1967 with a remarkable discoveryby Gardner et al. [5] of an exact method for solving the initial problem of the Korteweg de Vriesequation. In essence, they reduced the nonlinear problem to a linear one, which was well-knownas the Sturm-Liouville eigenvalue problem for the Schrodinger operator. They also discussedthe properties of the exact solution describing the interaction of solitons. This method, whichnow bears the name Inverse Scattering Transform, was later extended to a more general formto be applicable to a wide class of nonlinear evolution and wave equations such as the modifiedKdV equation, the nonlinear Schrodinger equation, the sine-Gordon equation, and many more

T-4162 2

[6, 7, 8, 9, 10].In 1971, Hirota [11] developed an ingenious method for obtaining the exact multi-soliton

solutions of the KdV equation and derived an explicit expression for its N -soliton solution. Anelegant formulation of this method requires the use of bilinear operators, therefore it is calledHirota’s bilinear method. Over the last decade this method has been shown to be applicableto a large class of nonlinear evolution equations, including difference-differential and integro-differential equations [12].

Hirota’s bilinear method, which is usually applied to completely integrable systems, is wellsuited for partially integrable equations as well. One can indeed conjecture that all completelyintegrable nonlinear evolution equations can be put into bilinear form. The converse is nottrue: a bilinear form can also be constructed for many equations that are not integrable. Inparticular, it is not true that any nonlinear evolution equation that can be written in bilinearform, automatically has a N -soliton solution for any value of N . In fact, even the existence oftwo-soliton solutions is nontrivial for a completely general bilinear equation. Hence, there is aneed for a straightforward algorithm to test whether or not an equation in bilinear form admitsa N -soliton solution; and if so for which values of N . A well-documented search for bilinearequations admitting soliton solutions has been done by Ito [13] and Hietarinta [14, 15, 16, 17, 18].

Hirota’s bilinear method has been studied and used extensively. The fundamental ideabehind the method is to use some dependent variable transformation to put the nonlinearevolution equation in a form where the new unknown function appears bilinearly. Once thebilinear form of the equation is found, one introduces a formal perturbation expansion toconstruct its solution step by step. If soliton solutions exist this expansion will always truncateand the now finite series will lead to an exact solution.

The drawback of Hirota’s method is that it requires a great deal of elementary algebraand calculus. These straightforward calculations can easily be performed with any symbolicmanipulation program, such as MACSYMA, REDUCE, MATHEMATICA, SCRATCHPAD,MAPLE and DERIVE.

In this thesis, we present MACSYMA programs based on the algorithm of Hirota’s bilinearmethod to carry out the symbolic calculation of one, two and three soliton solutions for botha single bilinear equation and for a system of coupled bilinear equations. These symbolicprograms also allow testing to see if one, two, three and four soliton solutions exist, withouthaving to construct these solutions explicitly.

We also give some test results for well-known nonlinear PDEs such as the Korteweg-deVries, the Boussinesq, the Kadomtsev-Petviashvili, the Sawada-Kotera and the shallow waterwave equations [10, 14, 15, 18, 19].

With these programs in place one can now start a comprehensive search for nonlinearevolution and wave equations that admit soliton solutions and therefore have some or all of theremarkable properties shared by completely integrable nonlinear PDEs.

As of today, there are no other programs available that allow testing for the existence ofexact soliton solutions and allow their construction without direction by the user. æ

2. HIROTA’S BILINEAR METHOD

Hirota’s method has been one of the most successful direct techniques for constructing exactsolutions to various nonlinear partial differential equations from soliton theory. For a review ofother direct methods we refer to papers by Hereman [20] and Hereman and Takaoka [21].

The method also allows testing if a certain equation satisfies the necessary requirements toadmit solitary wave solutions and soliton solutions.

The drawback of Hirota’s method is that the bilinear form must be known. In other words,the technique applies to any equation that can be written in bilinear form, either as a singlebilinear equation or as a system of coupled bilinear equations.

Once the bilinear form is obtained the method becomes algorithmic. Nothing is neededbeyond calculus and algebra. The calculations however become very lengthy and involved,in particular for PDEs of high order or with highly nonlinear terms. The complexity of thecalculations also drastically increases with the type of soliton solution one desires to obtain.Single soliton solutions are easy to calculate, even by hand, two and three soliton solutions arebarely manageable by hand. Four soliton solutions are at the limit of what a symbolic programcan do. Once the form of the two and three soliton solutions are known, their structure revealsthe form of higher soliton solutions. Using mathematical induction one can then prove whetheror not the hypothesized general solution satisfies the equation. This kind of conjecturing andtesting could be greatly assisted with any symbolic manipulation program.

Needless to say, this type of calculation is very suitable for a large scale symbolic manipu-lation program, such as MACSYMA, MATHEMATICA and REDUCE.

Here we only give a synopsis of the Hirota method which has been discussed in great detailin a vast amount of literature on soliton theory [9, 10, 12, 22, 23].

In essence, Hirota’s method requires:

(i) a clever change of dependent variable,(ii) the introduction of a novel differential operator,(iii) a perturbation expansion to solve the resulting bilinear equation.

2.1 The f -Function

Motivated both by the form of the N−soliton solution for the KdV, known from e.g. the InverseScattering Transform Method [9], and by a transformation for the Burgers’ equation (i.e. theCole-Hopf transformation [9] that reduces the Burgers’ equation into the linear heat equation),in 1971 Hirota defined a new function f as follows

u(x, t) = 2∂2 ln f(x, t)

∂x2 . (2.1)

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This change of dependent variable is quite often revealed by Painleve analysis of the equation[24, 25, 26, 27]. In many books [2, 7, 8, 18] this function f is now called the τ function becausenot only does it serve as a means of generating soliton solutions, but it also plays a crucial rolein the theoretical framework behind Hirota’s method.

2.2 The Bilinear Operator

Hirota introduced the differential operator Dx, defined on ordered pairs of functions f(x) andg(x), as follows

Dx(f ·g) =

(∂

∂x− ∂

∂x′

)f(x) g(x′)

∣∣∣∣x′=x

. (2.2)

More generally, he defined

Dmx Dn

t (f ·g) =

(∂

∂x− ∂

∂x′

)m (∂

∂t− ∂

∂t′

)n

f(x, t) g(x′, t′)∣∣∣∣x′=x,t′=t

, (2.3)

for non-negative integers m and n. This type of differential operator is called a bilinear operator,due to the obvious linearity in both its arguments. Bilinear operators Dn

y , Dnz , etc. could be

defined in a similar way.Let us look at how the operators act on simple functions. The following properties for the

bilinear operators in (2.2) and (2.3) are easily verified:

Dmx (f ·1) =

∂mf

∂xm, (2.4)

Dmx (f ·g) = (−1)mDm

x (g·f) , (2.5)

Dmx (f ·f) = 0 , for m odd , (2.6)

Dmx Dn

t (ek1x−ω1t·ek2x−ω2t) = (k1 − k2)m(−ω1 + ω2)

ne(k1+k2)x−(ω1+ω2)t . (2.7)

In particular, the last property will be very useful in the calculation of soliton solutions. LetP (Dt, Dx) be a polynomial in Dt and Dx. Then it follows from (2.4) and (2.7) that

P (Dx, Dt)(ek1x−ω1t·ek2x−ω2t)=

P (k1−k2,−ω1+ω2)

P (k1+k2,−ω1−ω2)P (Dx, Dt)(e

(k1+k2)x−(ω1+ω2)t·1) . (2.8)

These properties will be used extensively in the computer implementation of the Hirota method.

2.3 The Bilinear Form of a Single Equation

We take the Korteweg-de Vries (KdV) equation [19],

ut + 6uux + u3x = 0 , (2.9)

as the leading example to outline the Hirota procedure for a single equation.Carrying out the dependent variable transformation

u(x, t) = 2∂2 ln f(x, t)

∂x2 , (2.10)

T-4162 5

and one integration with respect to x, allows us to replace (2.9) by

ffxt − fxft + ff4x − 4fxf3x + 3f 22x = 0 . (2.11)

This quadratic equation in f can then be written in bilinear form,

P (Dx, Dt)(f ·f)def= B(f ·f)

def=(DxDt + D4

x

)(f ·f) = 0 , (2.12)

where the new operator Dx and Dt are given in (2.2) and (2.3). P should be considered as apolynomial in its arguments, B abbreviates the bilinear operator for the KdV equation.

Let us briefly consider some other examples of nonlinear PDEs with their bilinear represen-tations.

For the Sawada-Kotera (SK) equation [19],

ut + 45u2ux + 15uxuxx + 15uu3x + u5x = 0 , (2.13)

the dependent variable transformation (2.10) and one integration with respect to x yields

P (Dx, Dt)(f ·f)def=(DxDt + D6

x

)(f ·f) = 0 . (2.14)

For the Kadomtsev-Petviashvili (KP) equation [19],

(ut + 6uux + u3x)x + 3u2y = 0 , (2.15)

analogously, one would obtain

P (Dx, Dt, Dy)(f ·f)def=(DxDt + D4

x + 3D2y

)(f ·f) = 0 , (2.16)

where Dy is defined in a similar way as Dx and Dt.The shallow water wave (SW) equation [19],

uxxt + 3uut − 3ux

∫ x

ut dx− 2ux − ut = 0 , (2.17)

can also be transformed into bilinear form as

P (Dx, Dt)(f ·f)def=(D3

xDt −D2x −DxDt

)(f ·f) = 0 . (2.18)

Many more single equations from soliton theory can be transformed into a single new equationin bilinear form [12]. But not all!

2.4 The Bilinear Form of a Coupled System

Let us give an example of a famous soliton equation for which all attempts to derive a singlebilinear form fail. To obtain a bilinear representation for the so-called modified Korteweg-deVries (mKdV) equation [15],

ut + 6u2ux + u3x = 0 , (2.19)

much more work is required and one will end up with a system of bilinear equations.

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Introducing a new function w by u = wx, (2.19) can be replaced by

wtx + 6w2xwxx + w4x = 0 . (2.20)

Next, we integrate with respect to x and we set the integration constant equal to zero. Thus,

wt + 2w3x + w3x = 0 . (2.21)

Changing the dependent variable in (2.21) according the transformation

w(x, t) = −2i arctan(

G

F

)(2.22)

leads to(F 2 + G2)[(D3

x + Dt)(G·F )]− 3Dx(G·F )[D2x(F ·F + G·G)] = 0 . (2.23)

Since we still have two arbitrary functions F and G we require that

(D3x + Dt)(G·F ) = 0 , (2.24)

D2x(F ·F + G·G) = 0 , (2.25)

or upon complex rotation via f = F + iG, g = F − iG, so that w = log(f/g), one obtains

P1(Dx, Dt)(f ·g)def= (D3

x + Dt)(f ·g) = 0 , (2.26)

P2(Dx, Dt)(f ·g)def= D2

x(f ·g) = 0 . (2.27)

Various other single equations and, of course, systems of equations are transformable in termsof a system of bilinear equations [9, 12, 13, 14, 15, 16, 17, 18].

2.5 Hirota’s Direct Method

Our task is now to solve the bilinear equation. To make matters clear, let us continue withthe KdV case in (2.12). Introducing a bookkeeping parameter ε, we look for a formal seriessolution

f = 1 +∞∑

n=1

εn fn , (2.28)

for unknown functions f1(x, t), f2(x, t), etc.. Substituting (2.28) into (2.12) and equating tozero the powers of ε, yields

O(ε0) : B(1·1) = 0 , (2.29)

O(ε1) : B(1·f1 + f1·1) = 0 , (2.30)

O(ε2) : B(1·f2 + f1·f1 + f2·1) = 0 , (2.31)

O(ε3) : B(1·f3 + f1·f2 + f2·f1 + f3·1) = 0 , (2.32)

O(ε4) : B(1·f4 + f1·f3 + f2·f2 + f3·f1 + f4·1) = 0 , (2.33)

O(εn) : B

n∑j=0

fj·fn−j

= 0 , with f0 = 1 . (2.34)

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Of course, (2.29) is trivially satisfied. This scheme is general whatever the explicit expressionof the bilinear operator B is. For the KdV equation the operator B is defined in (2.12).

It is well-known [9, 10] that if the original PDE admits a N -soliton solution then (2.28) willtruncate at level n = N , provided f1 is the sum of precisely N simple exponential terms. Inthe sense that the series truncates, the Hirota method is very different from any traditionalperturbation expansion where the series would contain infinitely many terms.

The simplest solution of the KdV equation is the one soliton solution (N = 1) generatedfrom

f1 = exp θ = exp(kx− ω + δ) ,

where k, ω and δ are constant, equation (2.30) determines the dispersion law,

ω = k3 , (2.35)

and (2.31) allows to set f2 = 0. Consequently we can take fi = 0 for i > 2. Let ε = 1 and wehave

f = 1 + f1 = 1 + exp θ = 1 + exp(kx− ωt + δ) .

Now, substituting f in (2.10) with (2.35), we obtain

u(x, t) = 2∂2 ln f(x, t)

∂x2 = 2

(ffxx − f 2

x

f 2

)

=1

2k2sech2 1

2(kx− k3t + δ) .

Denoting1

2k = K, we get

u = 2K2sech2K(x− 4K2t + δ) ,

i. e. the pulse shaped solitary wave solution of the KdV equation.To construct the two soliton solution (N = 2) of the KdV equation we start with

f1 = exp θ1 + exp θ2

= exp(k1x− ω1 + δ1) + exp(k2x− ω2 + δ2) ,

where ki, ωi and δi are constant and i = 1, 2, Again (2.30) determines the dispersion law

ωi = k3i , i = 1, 2 . (2.36)

The terms generated by B(f1·f1) in (2.31) justify the choice

f2 = a12 exp(θ1 + θ2)

= a12 exp [(k1 + k2) x− (ω1 + ω2) t + (δ1 + δ2)]

and (2.31) allows to calculate the constant a12. With (2.36) one then obtains

a12 =(k1 − k2)

2

(k1 + k2)2 . (2.37)

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Subsequently, (2.32) allows to verify that indeed f3 = 0, and (2.33) allows to take fi = 0 fori > 3. Set ε = 1 in (2.28) to get

f = 1 + exp θ1 + exp θ2 + a12 exp(θ1 + θ2) , (2.38)

Upon substitution of f in (2.10), with (2.36) and upon selecting

eδi =c2i

ki

ekix−ωit+∆i for i = 1, 2 ,

f =1

4fe−

12(θ1+θ2) where θi = kix− ωit + ∆i , for i = 1, 2 ,

we obtain

u(x, t) = 2∂2 ln f(x, t)

∂x2 = 2∂2 ln f(x, t)

∂x2 = u(x, t) .

If we now take c2i =

(k2 + k1

k2 − k1

)ki for i = 1, 2, then

f(x, t) =(

1

k2 − k1

)(k2 cosh

θ1

2cosh

θ2

2− k1 sinh

θ1

2sinh

θ2

2

),

and u(x, t) can be rewritten in a nice form as

u(x, t) = u(x, t) = 2∂2 ln f(x, t)

∂x2

=

(k2

2 − k21

2

) k22cosech2 θ2

2+ k2

1sech2 θ1

2

(k2 coth θ2

2− k1 tanh θ1

2)2

.

Consider the case of a three soliton solution (N = 3). Then,

f1 =3∑

i=1

exp(θi) =3∑

i=1

exp (ki x− ωi t + δi) , (2.39)

where ki, ωi and δi are constants. Whereas (2.30) determines the dispersion law for the KdVequation,

ωi = k3i , i = 1, 2, 3 . (2.40)

The terms generated by B(f1·f1) in (2.31) justify the choice

f2 = a12 exp(θ1 + θ2) + a13 exp(θ1 + θ3) + a23 exp(θ2 + θ3)

= a12 exp [(k1 + k2) x− (ω1 + ω2) t + δ1 + δ2]

+ a13 exp [(k1 + k3) x− (ω1 + ω3) t + δ1 + δ3]

+ a23 exp [(k2 + k3) x− (ω2 + ω3) t + δ2 + δ3] , (2.41)

and (2.31) allows the calculation of the constants a12, a13 and a23. With (2.40) one obtains

aij =(ki − kj)

2

(ki + kj)2 , i, j = 1, 2, 3 with i < j . (2.42)

T-4162 9

Then, B(f1·f2 + f2·f1) in (2.32) motivates the particular solution

f3 = b123 exp(θ1 + θ2 + θ3)

= b123 exp [(k1+k2+k3)x−(ω1+ω2+ω3)t+(δ1+δ2+δ3)] , (2.43)

and one calculates

b123 = a12 a13 a23 =(k1 − k2)

2 (k1 − k3)2 (k2 − k3)

2

(k1 + k2)2 (k1 + k3)

2 (k2 + k3)2 . (2.44)

Subsequently, (2.33) allows us to verify that indeed f4 = 0. In the sixth equation of the schemeB(f2·f3 + f3·f2) should equal zero in order to assure that f5 = 0. If so, it will be possible totake fi = 0 for i ≥ 6. Finally, setting ε = 1 in (2.28), we obtain

f = 1 + exp θ1 + exp θ2 + exp θ3

+ a12 exp(θ1 + θ2) + a13 exp(θ1 + θ3) + a23 exp(θ2 + θ3)

+ b123 exp(θ1 + θ2 + θ3) , (2.45)

which upon substitution in (2.10) generates the well-known three soliton solution of (2.9). Thereis no simple formula found for u(x, t) of the three soliton solution of KdV equation.

The construction of N -soliton solutions [9, 10, 14, 15, 16, 17, 18, 22, 23] with N ≥ 3 is tediousand the necessary algebraic simplifications and factorizations are bound to fail if carried outby hand. Hence the need for a symbolic program that relieves us of the elaborate calculations.Before we enter the arena of symbolic manipulation we will discuss the conditions under whichHirota’s method can be applied and would lead to exact soliton solutions.

æ

3. THE HIROTA CONDITIONS

In this chapter we will use Hirota’s method to investigate whether or not a bilinear equation(or system of bilinear equations) has the required properties to admit one, two or three soli-ton solutions. This type of testing inherently allows us to verify if an equation is completelyintegrable, provided the bilinear form is known. Completely integrable means that the In-verse Scattering Transformation Method would be applicable and that the equation could belinearized and therefore solved. The linear equations that result from the Inverse ScatteringTransform Method are integral equations of Gelfand-Levitan-Marchenko type. More detailsabout the connection between Hirota’s and the Inverse Scattering Transform methods can befound in [28, 29].

3.1 Hirota’s Conditions and Soliton Solution for a Single Equation

Consider any bilinear equation of the form

P (Dx, Dt)(f ·f) = 0 . (3.1)

To obtain a single soliton solution we take

f = 1 + eθ , (3.2)

withθ = kx− ωt + δ , (3.3)

and where k, ω and δ are constant parameters.This f is a solution of (3.1), provided the parameters k and ω satisfy the dispersion relation,

P (k,−ω) = 0 . (3.4)

It is obvious that the polynomial P in (3.1) must be even,

P (Dx, Dt) = P (−Dx,−Dt) , (3.5)

and without a constant term,P (0, 0) = 0 . (3.6)

A one and also a two-soliton solution can always be constructed for (3.1) if the conditions (3.5)and (3.6) are satisfied. To obtain a two-soliton solution we start with

f = 1 + eθ1 + eθ2 + a12eθ1+θ2 (3.7)

T-4162 11

whereθi = kix− ωit + δi , i = 1, 2 . (3.8)

For f to satisfy (3.1) the dispersion law (3.4) should hold for each pair (ki, ωi) and coefficienta12 should be chosen as

a12 = −P (k1 − k2,−ω1 + ω2)

P (k1 + k2,−ω1 − ω2). (3.9)

This technique of constructing a one and two-soliton solution works for all nonlinear equationsthat can be cast in the bilinear form (3.1). Apart from (3.5) and (3.6) there are no further con-straints on P . Furthermore, using the polynomial P gives a fast and elegant way of calculatingthe coefficients a12, without having to use the bilinear operator explicitly.

Let us now turn to the construction of the N -soliton solution and the related conditions forP .

When the ansatz (3.7) is extended to three or more solitons we get stringent conditions forthe polynomial P . For the N -soliton solution, Hirota [23] starts with a generalization of (3.7)and writes f in the form

f =∑

µ=0,1

exp

(N)∑i<j

Aijµiµj +N∑

i=1

µiθi

, (3.10)

where the θi are as in (3.8) for i = 1, 2, . . . , N . The summation∑

µ=0,1 is over all combinations of

µ1 = 0, 1, µ2 = 0, 1, . . . , µN = 0, 1. The summation∑(N)

i<j stands for all possible combinationsunder the condition 0 < i < j ≤ N . For example, for N = 3, (3.10) will be given by

f = 1 + eθ1 + eθ2 + eθ3 + a12eθ1+θ2 + a13e

θ1+θ3 + a23eθ2+θ3 + b123e

θ1+θ2+θ3 ,µ1 = 0 µ1 = 1 µ1 = 0 µ1 = 0 µ1 = 1 µ1 = 1 µ1 = 0 µ1 = 1µ2 = 0 µ2 = 0 µ2 = 1 µ2 = 0 µ2 = 1 µ2 = 0 µ2 = 1 µ2 = 1µ3 = 0 µ3 = 0 µ3 = 0 µ3 = 1 µ3 = 0 µ3 = 1 µ3 = 1 µ3 = 1

where under each term we gave the corresponding µ values and where

aij = exp Aij and b123 = a12a13a23 .

When (3.10) is substituted into (3.1) we get at first order in eθ the condition (3.4) for theconstants ki, ωi, i = 1, 2, . . . , N . The constants aij or equivalently Aij are determined atsecond order in eθ by the analog of (3.9),

aij = exp Aij = −P (ki − kj,−ωi + ωj)

P (ki + kj,−ωi − ωj), i < j . (3.11)

At higher orders, Hirota [23] obtained the conditions

S[P, n] =∑

σ=±1

P

(n∑

i=1

σiki,−n∑

i=1

σiωi

)

×(n)∏i<j

P (σiki − σjkj,−σiωi + σjωj)σiσj = 0 (3.12)

T-4162 12

for each n = 2, 3, . . . , N . Where∑

σ=±1 indicates the summation over all possible combinations

of σ1 = ±1, σ2 = ±1, . . . , σn = ±1 and∏(n)

i<j means the product over all possible combinationsof n elements under the condition i < j, and all ki, ωi subject to (3.4). Since it is essential tothe bilinear transformation method, the explicit proof of the N -soliton condition (3.12) is givenin Appendix A. For n = 2 the condition (3.12) is satisfied automatically provided that (3.5)holds. The first nontrivial condition (3.12) occurs for n = N = 3.Example:

For the KdV equation the bilinear form is (2.12), hence the polynomial is

P (Dx, Dt) = DxDt + D4x . (3.13)

So we can verify the conditions (3.5) and (3.6) immediately

P (−Dx,−Dt) = (−Dx)(−Dt) + (−Dx)4

= DxDt + D4x

= P (Dx, Dt) ,

P (0, 0) = 0 .

A two soliton solution can always be constructed, starting with

f = 1 + eθ1 + eθ2 + a12eθ1+θ2 ,

whereθi = kix− ωit + δi , i = 1, 2 .

LetP (ki,−ωi) = 0 , i = 1, 2 ,

orki(−ωi) + k4

i = 0 , i = 1, 2 ,

from which the dispersion relation of the KdV follows

ωi = k3i , i = 1, 2 . (3.14)

Furthermore, according to (3.11),

a12 = −P (k1 − k2,−ω1 + ω2)

P (k1 + k2,−ω1 − ω2)

= −(k1 − k2)(−ω1 + ω2) + (k1 − k2)4

(k1 + k2)(−ω1 − ω2) + (k1 + k2)4.

Substitution of the dispersion relation (3.14) and simplification yields

a12 = −(k1 − k2)(−k31 + k2

2) + (k1 + k2)4

(k1 + k2)(−k31 − k3

2) + (k1 + k2)4

= −(k1 − k2)(−k31 + k3

2 + (k1 − k2)3

(k1 + k2)(−k31 − k3

2 + (k1 + k2)3

T-4162 13

= −(k1 − k2)(−k31 + k3

2 + k31 − 3k2

1k2 + 3k1k22 − k3

2)

(k1 + k2)(−k31 − k3

2 + k31 + 3k2

1k2 + 3k1k22 + k3

2)

= −(k1 − k2)(−3k21k2 + 3k1k

22)

(k1 + k2)(3k21k2 + 3k1k2

2)

=(k1 − k2)

2

(k1 + k2)2.

For the three soliton solution condition (n = N = 3), we need to verify that S[P, 3] = 0,this means that

S[P, 3] =∑

σ=±1

P

(3∑

i=1

σiki,−3∑

i=1

σiωi

)

×(3)∏i<j

P (σiki − σjkj,−σiωi + σjωj)σiσj (3.15)

=∑

σ=±1

P (σ1k1 + σ2k2 + σ3k3,−σ1ω1 − σ2ω2 − σ3ω3)

× P (σ1k1 − σ2k2,−σ1ω1 + σ2ω2)σ1σ2

× P (σ1k1 − σ3k3,−σ1ω1 + σ3ω3)σ1σ3

× P (σ2k2 − σ3k3,−σ2ω2 + σ3ω3)σ2σ3

= P (k1 + k2 + k3,−ω1 − ω2 − ω3)P (k1 − k2,−ω1 + ω2)

× P (k1 − k3,−ω1 + ω3)P (k2 − k3,−ω2 + ω3)

+ P (k1 + k2 − k3,−ω1 − ω2 + ω3)P (k1 − k2,−ω1 + ω2)

× [−P (k1 + k3,−ω1 − ω3)][−P (k2 + k3,−ω2 − ω3)]

+ P (k1 − k2 + k3,−ω1 + ω2 − ω3)[−P (k1 + k2,−ω1 − ω2)]

× P (k1 − k3,−ω1 + ω3)[−P (−k2 − k3, ω2 + ω3)]

+ P (k1 − k2 − k3,−ω1 + ω2 + ω3)[−P (k1 + k2,−ω1 − ω2)]

× [−P (k1 + k3,−ω1 − ω3)]P (−k2 + k3, ω2 − ω3)

+ P (−k1 + k2 + k3, +ω1 − ω2 − ω3)[−P (−k1 − k2, ω1 + ω2)]

× [−P (−k1 − k3, ω1 + ω3)]P (k2 − k3,−ω2 + ω3)

+ P (−k1 + k2 − k3, ω1 − ω2 + ω3)[−P (−k1 − k2, ω1 + ω2)]

× P (−k1 + k3, ω1 − ω3)[−P (k2 + k3,−ω2 − ω3)]

+ P (−k1 − k2 + k3, +ω1 + ω2 − ω3)P (−k1 + k2, ω1 − ω2)

× [−P (−k1 − k3, ω1 + ω3)][−P (−k2 − k3, ω2 + ω3)]

+ P (−k1 − k2 − k3, ω1 + ω2 + ω3)P (−k1 + k2, ω1 − ω2)

× P (−k1 + k3, ω1 − ω3)P (−k2 + k3, ω2 − ω3),

should vanish identically irrespective of the values of all the ki’s and the ωi’s.One can use the MACSYMA program HIROTA SINGLE.MAX to check the three soliton

condition (3.15) for the KdV bilinear equation (2.12) and to construct the three soliton solution.Here we list the results: S[P, 3] = 0, and

f = 1 + eθ1 + eθ2 + eθ3 + a12 eθ1+θ2 + a13 eθ1+θ3 + a23 eθ2+θ3 + b123 eθ1+θ2+θ3 ,

T-4162 14

withθi = kix− ωit + δi , i = 1, 2, 3 ,

ωi = k3i , i = 1, 2, 3 ,

and with coefficients

aij =(ki − kj)

2

(ki + kj)2, i, j = 1, 2, 3 for i < j ,

b123 = a12 a13 a23

=(k1 − k2)

2(k1 − k3)2(k2 − k3)

2

(k1 + k2)2(k1 + k3)2(k2 + k3)2. (3.16)

3.2 Hirota’s Conditions and Soliton Solutions for Systems of Equations

Recall that the system of bilinear equations for the mKdV is given by

P1(Dx, Dt)(f ·g) = 0 , (3.17)

P2(Dx, Dt)(f ·g) = 0 , (3.18)

where P1 is an odd polynomial in Dx, Dt and P2 is even in Dx, Dt, but without constant term,i.e.

Pi(−Dx,−Dt) = (−1)iPi(Dx, Dt) , i = 1, 2 , (3.19)

P2(0, 0) = 0 . (3.20)

As soon as the above simple conditions are met, a one and two soliton solution can be obtained.For the one-soliton solution we take

f = 1 + eθ , (3.21)

g = 1− eθ , (3.22)

withθ = kx− ωt + δ . (3.23)

Substituting f and g into the coupled bilinear equations (3.17) and (3.18), it is easy to showthat k and ω must satisfy the dispersion relation

P1(k,−ω) = 0 . (3.24)

A two soliton solution can always be constructed via the obvious generalization

f = 1 + eθ1 + eθ2 + a12eθ1+θ2 , (3.25)

g = 1− eθ1 − eθ2 + a′12eθ1+θ2 , (3.26)

whereθi = kix− ωit + δi , i = 1, 2 , (3.27)

T-4162 15

andP1(ki,−ωi) = 0 , i = 1, 2 . (3.28)

We substituted (3.25) and (3.26) into (3.17) and (3.18) and also used (3.27) and (3.28). Wefound that (3.25) and (3.26) will be a solution of (3.17) and (3.18) if a12 and a′12 are given by

a′12 = a12 =P2(k1 − k2,−ω1 + ω2)

P2(k1 + k2,−ω1 − ω2). (3.29)

In [18] it is shown that the N -soliton solution of the system (3.17) and (3.18), will follow from

f =∑

µ=0,1

exp

(N)∑i<j

Aijµiµj +N∑

i=1

µi(θi +iπ

2)

, (3.30)

g =∑

µ=0,1

exp

(N)∑i<j

Aijµiµj +N∑

i=1

µi(θi −iπ

2)

, (3.31)

where the θi are given in (3.27). The parameters ki and ωi must again satisfy the dispersionrelation (3.28) while the Aij are determined as in (3.29),

aij = exp Aij =P2(ki − kj,−ωi + ωj)

P2(ki + kj,−ωi − ωj), i < j . (3.32)

At higher orders (N ≥ 3), there are two conditions [18] for the polynomials Pi, here i = 1, 2,namely,

Sodd[P1, P2, n] =∑

σ=±1

P1

(n∑

i=1

σiki,−n∑

i=1

σiωi

)sin

(n∑

i=1

σiπ

2

)

×(n)∏i<j

P2(σiki − σjkj,−σiωi + σjωj) = 0 , (3.33)

Seven[P2, n] =∑

σ=±1

P2

(n∑

i=1

σiki,−n∑

i=1

σiωi

)cos

(n∑

i=1

σiπ

2

)

×(n)∏i<j

P2(σiki − σjkj,−σiωi + σjωj) = 0 , (3.34)

for each n = 2, 3, . . . , N and all ki, ωi subject to (3.28). The proof of the N -soliton conditionsfor a coupled equations is similar to the proof of the N -soliton condition for a single equationas given in Appendix A.

æ

4. MACSYMA PROGRAMS FOR THE HIROTA METHOD

In this Chapter we discuss some of the features used in the symbolic programs HIROTA SINGLE.MAXand HIROTA SYSTEM.MAX that carry out the calculations for Hirota’s algorithm to constructmulti-soliton solutions.

The symbolic programs calculates the one, two and three soliton solutions of a fairly simplePDE. The basic requirement is that the PDE can be transformed into either a single bilinearequation for the new variable f or a system of two coupled bilinear equations for new variablesf and g. Even for the latter type some restrictions apply: the coupled system must be of mKdVtype.

The current version of the program HIROTA SYSTEM.MAX can not handle equationssuch as the nonlinear Schrodinger equation, the sine-Gordon equation, etc., which all lead todifferent types of coupled systems. The development of a computer program that calculatessoliton solutions of an entire family of coupled bilinear systems would make an interesting topicfor further study.

The programs are written in such a way that the extension for the N -soliton is straightfor-ward. The structure of the programs should also allow to ‘translate’ them into the languagesof e.g. MATHEMATICA, MAPLE or REDUCE.

The programs require little interaction from the user, who must provide the bilinear op-erator(s) for the original PDE and specify which soliton solution (one, two or three) shouldbe calculated. The user can also control the execution of the condition and comparision testsdiscussed below.

A flowchart of the program HIROTA SINGLE.MAX, i.e. the MACSYMA program for a sin-gle bilinear equation, is given in Appendix B. The flowchart for the program HIROTA SYSTEM.MAXis similar.

4.1 Special Features Used in the Programs

There is no need to use the definitions (2.2) and (2.3) for the Hirota operators Dnx and Dm

x Dnt .

Instead one can use the following equivalent but much simpler expressions:

Dnx(f ·g) =

n∑j=0

(−1)(n−j)n!

j!(n− j)!

∂jf

∂xj

∂n−jg

∂xn−j, (4.1)

Dmx Dn

t (f ·g) =m∑

j=0

n∑i=0

(−1)(m+n−j−i)m!

j!(m− j)!

n!

i!(n− i)!

∂i+jf

∂ti∂xj

∂n+m−i−jg

∂tn−i∂xm−j. (4.2)

The observant reader will recognize these as the Leibniz formula for the derivatives ofproducts, up to an alteration in signs. We show in Appendix A that the above formulae areequivalent with Hirota’s bilinear operators defined in Chapter 2.

T-4162 17

Nowhere in the program do we use the explicit forms of exponentials. Instead we introduce

the functions h(x, t) with the properties∂h(x, t)

∂x= kh(x, t) and

∂h(x, t)

∂t= −ωh(x, t). These

partial derivatives will be assigned to the function h via the gradef command in MACSYMA.This trick avoids long expressions involving sums and products of exponentials.

The implementation of Hirota’s perturbation scheme given in (2.29)-(2.34) is nontrivial.Due to memory and space limitations, terms of (2.32) have to be calculated separately andadded up after necessary simplifications. In other words, MACSYMA can not handle thelengthy expressions resulting from applying the bilinear operator on sums of pairs of functionsoccurring in (2.32).

For complicated equations MACSYMA can no longer handle all the terms needed to testthe existence of a four soliton solution. For example, for the KdV equation the test for theexistence of a three soliton solution involves verification that the sum of 8× 52× 3× 9 = 11232terms vanishes. Similarly for a four soliton solution to exist the algorithm requires checkingwhether or not the sum of 16× 52× 6× 9 = 44928 terms vanishes. The computer (VAX 8600)can still handle that case! For the same reason only the test for the existence of a three andfour soliton solution are included in the programs. More efficient ways to test the existence ofsolitons are under investigation.

4.2 The Algorithm of HIROTA SINGLE.MAX

There are 14 blocks (functions) in the program HIROTA SINGLE.MAX.We will discuss them briefly.

• Block 1: commentinter(name).This block gives a friendly interface.

• Block 2: dispersion(P,check coefficients,n,name).It calculates the dispersion relation for each of the wave numbers ki and angular frequen-cies ωi via the polynomial P , i.e. P (k,−ω) = 0.

• Block 3: condition 4soliton(P).It tests the condition for the existence of a four soliton solution based on Hirota’s conditionS[P, 4]. If the polynomial P satisfies the condition, then the block returns TRUE elseFALSE.

• Block 4: condition 3soliton(P).It tests the condition for the existence of a three soliton solution via Hirota’s conditionS[P, 3]. If the polynomial P satisfies the condition, then the block returns TRUE elseFALSE.

• Block 5: condition 1 2soliton(P).It tests the condition for the existence of one and two soliton solutions via Hirota’scondition, i.e. it verifies whether P is an even function without constant term. If thepolynomial P satisfies these conditions, then the block returns TRUE else FALSE.

T-4162 18

• Block 6: construct 2soliton(B,n).This block constructs the function f2 and finds the coefficient aij via Hirota’s bilinearoperator.

• Block 7: construct 3soliton(B,n).This block constructs the function f3 and finds the coefficient b123 via Hirota’s bilinearoperator.

• Block 8: a ij(P).This block calculates the coefficient aij via the polynomial form.

• Block 9: check a().This block compares the coefficients aij which are now calculated by the two methods. Ifthe coefficients are equal, it returns TRUE else FALSE.

• Block 10: b 123(P).This block calculates the coefficient b123 via the polynomial form.

• Block 11: check b().This block verifies equality of the coefficients b123 which are now calculated with the twodifferent methods. If the coefficients are the same, its returns TRUE else FALSE.

• Block 12: hirota op(B).This block defines the Hirota’s operators Dn

x and Dmx Dn

t by (4.1) and (4.2). The operatorDn

y is also defined in this block and furthermore this block determines the polynomial Pcorresponding to the bilinear operator B.

• Block 13: hirota(B,name,N,test for 3soliton,check coefficients, test for 4soliton).This block is the main program. B stands for the bilinear operator B(f, g) for the givenPDE.The user can assign a ‘name’ to the PDE, e.g. name: Korteweg de Vries equation.The N refers to the N -soliton solution the user wants to calculate. N is either one, twoor three.If test for 3soliton is TRUE then Hirota’s condition for the existence of the three solitonsolution will be tested. If test for 3soliton is set to FALSE then that test will be skipped.One sets check coefficients to TRUE or FALSE. If TRUE, the program will verify whetheror not the coefficients aij and b123, which are calculated first via the polynomial form,coincide with the corresponding coefficients obtained via Hirota’s bilinear operator.

• Block 14: output(N).This block outputs the function f for the N -soliton solution. The solution u(x, t) ofthe original PDE is then obtained via the transformation formula relating f and u. For

instance, for the KdV equation one has u(x, t) = 2∂2 ln f(x, t)

∂x2 .

T-4162 19

4.3 The Batch Program for HIROTA SINGLE.MAX

In a batch file the user must provide the bilinear operator B, give a name for the PDE, select thevalue of N (either 1, 2 or 3), and set the Boolean variables test for 3soliton, check coefficientsand test for 4soliton to TRUE or FALSE.

Obviously the verification of coefficients is only relevant for N > 1, so there are twentypossible logical combinations, as given in Table C.1 in Appendix C.As an example we give the batch program we used for the construction of the three solitonsolution for the KdV equation.

writefile(”test kdv.out”)$loadfile(”hirota single.lsp”)$N:3$B(f,g):=Dxt[1,1](f,g)+Dx[4](f,g)$name:Korteweg de Vries$hirota(B,name,N,true,true,true)$closefile()$quit()$

We decided to compile the program HIROTA SINGLE.MAX to improve the speed. The com-piled version HIROTA SINGLE.LSP is obtained via the save command in MACSYMA (seethe MACSYMA manual [30]).

4.4 The Algorithm of HIROTA SYSTEM.MAX

There are 14 blocks (functions) in the program HIROTA SYSTEM.MAX.

• Block 1: commentinter(name).This block gives a friendly interface.

• Block 2: dispersion(P1,check coefficients,n,name).It calculates the dispersion relation for each of the wave numbers ki and angular frequen-cies ωi via the polynomial P1, i.e. P1(k,−ω) = 0.

• Block 3: condition 4soliton(P1,P2).It tests the condition for the existence of a four soliton solution via Hirota’s conditionsSeven[P2, 4] and Sodd[P1, P2, 4]. If the polynomials P1 and P2 satisfy the conditions, thenthe block returns TRUE else FALSE.

• Block 4: condition 3soliton(P1,P2).It tests the condition for the existence of a three soliton solution by Hirota’s conditionsSeven[P2, 3] and Sodd[P1, P2, 3]. If the polynomials P1 and P2 satisfy the conditions, thenthe block returns TRUE else FALSE.

• Block 5: condition 1 2soliton(P1,P2).This block tests the conditions for the existence of the one and two soliton solutionsvia Hirota’s condition, i.e. it simply verifies whether P1 is odd function and P2 is even

T-4162 20

function without constant term. If the polynomials P1 and P2 satisfy these conditions,then the block returns TRUE else FALSE.

• Block 6: construct 2soliton(B2,n).This block constructs the functions f2 and g2 and finds the coefficient aij via Hirota’sbilinear operator.

• Block 7: construct 3soliton(B1,n). This block constructs the functions f3 and g3 andfinds the coefficient b123 via Hirota’s bilinear operator.

• Block 8: a ij(P2).This block calculates the coefficient aij via the polynomial form.

• Block 9: check a().This block compares the coefficients aij which are now calculated by the two methods. Ifthe coefficients are equal, it returns TRUE else FALSE.

• Block 10: b 123(P2).This block calculates the coefficient b123 via the polynomial form.

• Block 11: check b().This block verifies equality of the coefficients b123 which are now calculated via two meth-ods. If the coefficients are the same, it returns TRUE else FALSE.

• Block 12: hirota op(B1,B2).This block defines the Hirota’s operators Dn

x and Dmx Dn

t by (4.1) and (4.2). The operatorDn

y is also defined in this block and furthermore it determines the polynomials P1 and P2

corresponding to the bilinear operators B1 and B2.

• Block 13:hirota(B1,B2,name,N,test for 3soliton,check coefficients, test for 4soliton).This block is the main program. The symbols B1 and B2 stand for the bilinear operatorsB1(f, g) and B2(f, g) for the PDE. The operator B1 should be an odd operator and B2should be an even operator without constant term.The user can give a ’name’ to the PDE, e.g. name: modified Korteweg de Vries equation.The N refers to the N -soliton solution the user wants to calculate. N is either one, twoor three.If test for 3soliton is TRUE then Hirota’s conditions for the existence of the three solitonsolution will be tested. If test for 3soliton is FALSE then that test will be skipped.One sets check coefficients to TRUE or FALSE. If TRUE, the program will verify whetheror not the coefficients aij and b123, which are calculated first via the polynomial form,coincide with the corresponding coefficients obtained via Hirota’s bilinear operators.

• Block 14: output(N).This block returns the functions f and g for the N -soliton solution.

T-4162 21

4.5 The Batch Program for HIROTA SYSTEM.MAX

For the coupled system, the batch program is similar to the one for a single equation. But theuser must provide two bilinear operators B1 and B2, the first one must be odd, the second oneeven without constant term.Here is the batch program we used for the construction of the 3-soliton solution for the mKdVequation.

writefile(”test mkdv.out”)$loadfile(”hirota system.lsp”)$N:3$B1(f,g):=Dxt[0,1](f,g)+Dx[3](f,g)$B2(f,g):=Dx[2](f,g)$name:modified Korteweg de Vries$hirota(B1,B2,name,N,true,true,true)$closefile()$quit()$æ

5. EXAMPLES AND TEST CASES

The programs HIROTA SINGLE.MAX and HIROTA SYSTEM.MAX have been thoroughlytested. We did a search in the literature to find soliton equations that were known to admitthree soliton solutions. We also tested some equations that do not satisfy the criteria foradmitting a two and/or three soliton solution.

In this Chapter we give the results for the test examples. Unless stated otherwise we selectedN = 3, and the variables test for 3soliton, check coefficients and test for 4soliton were set toTRUE. For some of the examples the actual computer output is listed in Appendix D.

5.1 Test Results for Single Equations

We used the program HIROTA SINGLE.MAX to test the examples below. For one solitonsolutions the function f is

f = 1 + exp θ ,

for two soliton solutions the function f is given by

f = 1 + exp θ1 + exp θ2 + a12 exp(θ1 + θ2) ,

for three soliton solutions one has

f = 1 + exp θ1 + exp θ2 + exp θ3

+ a12 exp(θ1 + θ2) + a13 exp(θ1 + θ3) + a23 exp(θ2 + θ3)

+ b123 exp(θ1 + θ2 + θ3) .

In all the cases we calculated the coefficients aij and b123 in two different ways, via the polyno-mial form and via the bilinear operator, and found that the coefficients were equal.

• For the KdV equation (2.9),ut + 6uux + u3x = 0 ,

the bilinear operator [14, 19] is

B(f, g) = Dxt[1, 1](f, g) + Dx[4](f, g)

There exists a four soliton solution. In the expansion we use θi = kix−ωit+δi, where thewave number and the angular frequency depend on each other via the dispersion relationω = k3. The coefficients of the three soliton solution are

aij =(ki − kj)

2

(ki + kj)2 , i, j = 1, 2, 3 and i < j ,

T-4162 23

and

b123 = a12 a13 a23 =(k1 − k2)

2 (k1 − k3)2 (k2 − k3)

2

(k1 + k2)2 (k1 + k3)

2 (k2 + k3)2 .

The output of the program confirmed the results in (2.40), (2.42) and (2.44).

• For the Sawada-Kotera equation (SK) (2.13),

ut + 45u2ux + 15uxu2x + 15uu3x + u5x = 0 ,

the bilinear operator [19] is

B(f, g) = Dxt[1, 1](f, g) + Dx[6](f, g) .

There is a four soliton solution. In the expansion θi = kix − ωit + δi. The dispersionrelation is given by ω = k5. The coefficients in the three soliton solution are

aij =(ki − kj)

2 (k2i − kikj + k2

j )

(ki + kj)2(k2

i + kikj + k2j )

=(ki − kj)

3 (k3i + k3

j )

(ki + kj)3 (k3

i − k3j )

, i, j = 1, 2, 3 and i < j ,

b123 = a12 a13 a23

=(k1 − k2)

3 (k31 + k3

2)(k1 − k3)3 (k3

1 + k33)(k2 − k3)

3 (k32 + k3

3)

(k1 + k2)3 (k3

1 − k32)(k1 + k3)

3 (k31 − k3

3)(k2 + k3)3 (k3

2 − k33)

.

• For the seventh-order KdV equation, the bilinear operator for equation (b·1) in [13] withn = 3 is

B(f, g) = Dxt[1, 1](f, g) + Dx[8](f, g) .

There is a two soliton solution but no three soliton solution. In the expansion θi =kix− ωit + δi with ω = k7. The coefficient of the two soliton solution is

a12 =(k1 − k2)

2(k21 − k1k2 + k2

2)

(k1 + k2)2(k21 + k1k2 + k2

2).

• For the modified fifth-order KdV equation, the bilinear operator of equation (b·2) in [13]is

B(f, g) = Dxt[3, 1](f, g) + Dx[6](f, g) .

There is a two soliton solution but no three soliton solution. In the expansion θi =kix − ωit + δi. The dispersion relation is ω = k3 and the coefficient in the two solitonsolution is

a12 =(k1 − k2)

4

(k1 + k2)4.

T-4162 24

• For the bilinear operator (b·4) in [13],

B(f, g) = Dxt[1, 1](f, g) + Dxt[2, 2](f, g) ,

there is a two soliton solution but no three soliton solution. In the expansion θi =

kix − ωit + δi and the dispersion relation is ω =1

k. The coefficient of the two soliton

solution is

a12 = −(k1 − k2)2(k2

1 − k1k2 + k22)

(k1 + k2)2(k21 + k1k2 + k2

2).

• For the bilinear operator (b·6) in [13],

B(f, g) = Dxt[0, 2](f, g) + Dxt[3, 1](f, g)∓Dy[2](f, g) ,

there exists a two soliton solution but no three soliton solution. In the expansion θi =

kix− ωit + liy + δi. The dispersion relation is ω=−√

k6±4l2−k3

2. The coefficient of the

two soliton solution is

a12=3k1k2(k2K2−k1K2−k2K1+k1K1−k4

2−k41+k1k

32+k3

1k2)+K1K2−k31k

32∓4l1l2

3k1k2(k2K2+k1K2+k2K1+k1K1−k42−k4

1−k1k32−k3

1k2)+K1K2−k31k

32∓4l1l2

,

where we used the abbreviations Ki =√

k6i ±4l2i , i = 1, 2 .

• For the bilinear operator (b·8) in [13],

B(f, g) = Dx[2](f, g)−Dxt[0, 2](f, g)−Dxt[2, 2](f, g) ,

there exists a two soliton solution but no three soliton solution. In the expansion θi =

kix−ωit+ δi with the dispersion relation ω=− 1√k2+1

. The coefficient of the two soliton

solution is given by

a12=2k2

2K1K2−4k1k2K1K2+2k21K1K2+2K1K2−k1k

32+2k2

1k22−k3

1k2−2k1k2−2

2k22K1K2+4k1k2K1K2+2k2

1K1K2+2K1K2+k1k32+2k2

1k22+k3

1k2+2k1k2−2,

where Ki =√

k2i +1, i = 1, 2 .

• For the bilinear operator (b·9) in [13],

B(f, g) = Dxt[0, 2](f, g) + Dx[6](f, g) ,

there exists a two soliton solution but no three soliton solution. In the expansion θi =kix − ωit + δi and the dispersion relation is ω = −ik3 with i =

√−1. The coefficient of

the two soliton solution is

a12 =(k1 − k2)

2(2k21 − k1k2 + 2k2

2)

(k1 + k2)2(2k21 + k1k2 + 2k2

2).

T-4162 25

• For the Kadomtsev-Petviashvili equation (2.15),

(ut + 6uux + u3x)x + 3u2y = 0 ,

the bilinear operator [18, 19] reads

B(f, g) = Dxt[1, 1](f, g) + Dx[4](f, g) + 3∗Dy[2](f, g) .

In this case we got a result only for test for 4soliton set to FALSE, due to space andmemory limitations of MACSYMA on a VAX 8600. There is a three soliton solution. In

this case θi = kix + liy−ωit. We obtain the dispersion relation ω =3l2 + k4

k, and for the

coefficients of the three soliton solution:

aij =(kilj − kik

2j − likj + k2

i kj)(kilj + kik2j − likj − k2

i kj)

(kilj − kik2j − likj − k2

i kj)(kilj + kik2j − likj + k2

i kj), i, j =1, 2, 3 , i<j ,

andb123 = a12 a13 a23 .

• For the Boussinesq equation [19, 23]

u2t − u2x − 3(u2)2x − u4x = 0 ,

the bilinear operator is

B(f, g) = Dxt[0, 2](f, g)−Dx[2](f, g)−Dx[4](f, g) .

There is a four soliton solution. In the expansion we use θi = kix−ωit+δi. The dispersionrelation is given by ω = −k

√1 + k2. The coefficients in the three soliton solution are

aij =

√1 + k2

i

√1 + k2

j − 2k2i + 3kikj − 2k2

j − 1√1 + k2

i

√1 + k2

j − 2k2i − 3kikj − 2k2

j − 1, i, j = 1, 2, 3 and i < j ,

andb123 = a12 a13 a23 .

In this case we got a result only for check coefficients set to FALSE, partly due to the factthat the dispersion relation is irrational and partly due to space and memory limitationsof MACSYMA on a VAX 8600. So, we only calculated the coefficients aij and b123 via thepolynomial form. One way to remove the square root in a formula is by a uniformizationtransformation. The procedure is as follows: in the dispersion relation

ω = −k√

1 + k2 ,

one puts

k =1

2

(r − 1

r

).

T-4162 26

Then

ω =1

4

(1

r2− r2

).

We have not yet implemented this option into our programs. The goal is to have thisuniformization process in an automated form. Should the program detect the appearanceof a

√in the dispersion law, it should warn the user that it will try to perform a

uniformization transformation or request a suitable substitution if it does not succeedfinding one by itself.

• For the shallow water wave equation [19],

uxxt + 3uut − 3ux

∫xut dx′ − ux − ut = 0 ,

the bilinear operator is

B(f, g) = Dxt[3, 1](f, g)−Dx[2](f, g)−Dxt[1, 1](f, g) .

In this case we got a result only for test for 4soliton set to FALSE, due to space andmemory limitations of MACSYMA on a VAX 8600. There is a three soliton solution. In

this case θi = kix − ωit + δi. We obtain the dispersion relation ω =k

(1 + k)(1− k)and

for the coefficients of the three soliton solution:

aij =(ki − kj)

2(k2i − kikj + k2

j − 3)

(ki + kj)2(k2i + kikj + k2

j − 3), i, j = 1, 2, 3 and i < j ,

b123 = a12 a13 a23

=(k1−k2)

2K12(k1−k3)2K13(k2−k3)

2K23

(k1+k2)2L12(k1+k3)2L13(k2+k3)2L23

,

where we used the abbreviations Kij = k2i − kikj + k2

j − 3 and Lij = k2i + kikj + k2

j − 3 ,for i, j = 1, 2, 3 with i < j.

5.2 Test Results for Coupled Systems

For all the cases below, the two soliton solution is generated by

f = 1 + i exp θ1 + i exp θ2 − a12 exp(θ1 + θ2) ,

g = 1− i exp θ1 − i exp θ2 − a12 exp(θ1 + θ2) , (5.1)

and the three soliton solution by

f = 1 + i exp θ1 + i exp θ2 + i exp θ3

− a12 exp(θ1 + θ2)− a13 exp(θ1 + θ3)− a23 exp(θ2 + θ3)

− ib123 exp(θ1 + θ2 + θ3) ,

g = 1− i exp θ1 − i exp θ2 − i exp θ3

− a12 exp(θ1 + θ2)− a13 exp(θ1 + θ3)− a23 exp(θ2 + θ3)

+ i b123 exp(θ1 + θ2 + θ3) .

For all cases tested, the coefficients aij and b123 calculated via both methods were the same.

T-4162 27

• For the modified Korteweg-de Vries equation (2.19),

vt + 6v2vx + v3x = 0 ,

the bilinear system [15] is

B1(f, g) = Dxt[0, 1](f, g) + Dx[3](f, g) ,

B2(f, g) = Dx[2](f, g) .

There is a four soliton solution. In the expansion θi = kix − ωit + δi and the dispersionrelation is ω = k3 . The coefficients of the three soliton solution are

aij =(ki − kj)

2

(ki + kj)2 , i, j = 1, 2, 3 and i < j ,

and

b123 = a12 a13 a23 =(k1 − k2)

2 (k1 − k3)2 (k2 − k3)

2

(k1 + k2)2 (k1 + k3)

2 (k2 + k3)2 .

• For the fifth-order modified Korteweg-de Vries equation, the bilinear system (b·11) in [13]with n = 2 reads

B1(f, g) = Dxt[0, 1](f, g) + Dx[5](f, g) ,

B2(f, g) = Dx[2](f, g) ,

and there is a four soliton solution. In the expansion θi = kix− ωit + δi. The dispersionrelation is ω = k5 and the coefficients of the three soliton solution are

aij =(ki − kj)

2

(ki + kj)2 , i, j = 1, 2, 3 and i < j ,

and

b123 = a12 a13 a23 =(k1 − k2)

2 (k1 − k3)2 (k2 − k3)

2

(k1 + k2)2 (k1 + k3)

2 (k2 + k3)2 .

• For the seventh-order modified Korteweg-de Vries equation, the bilinear system (b·11) in[13] with n = 3 is

B1(f, g) = Dxt[0, 1](f, g) + Dx[7](f, g) ,

B2(f, g) = Dx[2](f, g) ,

and there exists a four soliton solution. In the expansion θi = kix − ωit + δi and thedispersion relation reads ω = k7. The coefficients of the three soliton solution are

aij =(ki − kj)

2

(ki + kj)2 , i, j = 1, 2, 3 and i < j ,

and

b123 = a12 a13 a23 =(k1 − k2)

2 (k1 − k3)2 (k2 − k3)

2

(k1 + k2)2 (k1 + k3)

2 (k2 + k3)2 .

T-4162 28

• For the ninth-order modified Korteweg-de Vries equation, the bilinear system (b·11) in[13] with n = 4 is

B1(f, g) = Dxt[0, 1](f, g) + Dx[9](f, g) ,

B2(f, g) = Dx[2](f, g) ,

and the program confirms that there is a two soliton solution but no three soliton solution.In the expansion θi = kix − ωit + δi where ω = k9 . The coefficient of the two solitonsolution is

a12 =(k1 − k2)

2

(k1 + k2)2 .

• For yet another type of a high-order modified Korteweg-de Vries equation, the bilinearsystem (b·12) in [13] is

B1(f, g) = Dxt[0, 1](f, g) + Dx[3](f, g) ,

B2(f, g) = Dx[4](f, g) .

There is a two soliton solution but no three soliton solution. In the expansion θi =kix − ωit + δi and the dispersion relation is given by ω = k3. The coefficient of the twosoliton solution is

a12 =(k1 − k2)

4

(k1 + k2)4 .

• For the Backlund transformation associated with the equation that models shallow waterwaves, the bilinear system (b·13) in [13] is

B1(f, g) = Dxt[0, 1](f, g) + Dx[1](f, g)−Dx[2, 1](f, g) ,

B2(f, g) = Dx[2](f, g) .

There is a four soliton solution. In the expansion θi = kix− ωit + δi with ω = − k

k2 − 1.

The coefficients of the three soliton solution are

aij =(ki − kj)

2

(ki + kj)2 , i, j = 1, 2, 3 and i < j ,

and

b123 = a12 a13 a23 =(k1 − k2)

2 (k1 − k3)2 (k2 − k3)

2

(k1 + k2)2 (k1 + k3)

2 (k2 + k3)2 .

• For the bilinear system (b·14) in [13] ,

B1(f, g) = Dxt[0, 1](f, g) + Dx[1](f, g)−Dxt[2, 1](f, g) ,

B2(f, g) = Dxt[1, 1](f, g) ,

T-4162 29

there is a four soliton solution. In the expansion θi = kix − ωit + δi with the dispersion

relation ω = − k

k2 − 1. The coefficients of the three soliton solution are

aij =(ki − kj)

2(kikj + 1)

(ki + kj)2(kikj − 1), i, j = 1, 2, 3 and i < j ,

and

b123 = a12 a13 a23

=(k1 − k2)

2 (k1 − k3)2 (k2 − k3)

2 (k1k2 + 1) (k1k3 + 1) (k2k3 + 1)

(k1 + k2)2 (k1 + k3)2 (k2 + k3)2 (k1k2 − 1) (k1k3 − 1) (k2k3 − 1).

• For the bilinear system (b·15) in [13] with n = 1 ,

B1(f, g) = Dxt[0, 1](f, g) + Dx[3](f, g) ,

B2(f, g) = Dxt[1, 1](f, g) + Dx[4](f, g) ,

there is a four soliton solution. In the expansion θi = kix − ωit + δi. The dispersionrelation is ω = k3 and the coefficients are

aij = −(ki − kj)2

(ki + kj)2 , i, j = 1, 2, 3 and i < j ,

and

b123 = −(k1 − k2)2 (k1 − k3)

2 (k2 − k3)2

(k1 + k2)2 (k1 + k3)

2 (k2 + k3)2 .

• For the bilinear system (b·15) in [13] with n = 2 ,

B1(f, g) = Dxt[0, 1](f, g) + Dx[5](f, g) ,

B2(f, g) = Dxt[1, 1](f, g) + Dx[6](f, g) ,

there exists a two soliton solution but there is no three soliton solution. In the expansionθi = kix− ωit + δi. The dispersion relation is ω = k5 and the two soliton solution has thecoefficient

a12 =(k1 − k2)

2(k21 − k1k2 + k2

2)

(k1 + k2)2(k21 + k1k2 + k2

2).

• For the bilinear system (b·16) in [13] with n = 1 ,

B1(f, g) = Dxt[0, 1](f, g) + Dx[3](f, g) ,

B2(f, g) = Dxt[0, 2](f, g) + Dxt[3, 1](f, g) ,

there is a four soliton solution. In the expansion θi = kix − ωit + δi with ω = k3. Thecoefficients of the three soliton solution are

aij = −(ki − kj)

2(k2i + kikj + k2

j )

(ki + kj)2(k2i − kikj + k2

j ), i, j = 1, 2, 3 and i < j ,

T-4162 30

and

b123 =−(k1 − k2)2(k1 − k3)

2 (k2 − k3)2K12K13K23

(k1 + k2)2(k1 + k3)2(k2 + k3)2L12L13L23

.

Where we used the abbreviations Kij = k2i + kikj + k2

j and Lij = k2i − kikj + k2

j , fori, j = 1, 2, 3 with i < j .

• For the bilinear system (b·16) in [13] with n = 2 ,

B1(f, g) = Dxt[0, 1](f, g) + Dx[5](f, g) ,

B2(f, g) = Dxt[0, 2](f, g) + Dxt[5, 1](f, g) ,

there exists a two soliton solution but there is no three soliton solution. In the expansionθi = kix− ωit + δi with ω = k5. The coefficient of the two soliton solution is

a12 =(k1 − k2)

2(k21 − k1k2 + k2

2)(k41 + k3

1k2 + k21k

22 + k1k

32 + k4

2)

(k1 + k2)2(k21 + k1k2 + k2

2)(k41 − k3

1k2 + k21k

22 − k1k3

2 + k42)

.

• For the bilinear system with a parameter a ,

B1(f, g) = Dxt[0, 1](f, g) + Dx[3](f, g) ,

B2(f, g) = Dxt[1, 1](f, g) + a ∗Dx[4](f, g) ,

there exists a two soliton solution. For a three soliton solution to exist,

72(a− 1)a(2a + 1)k1k2k3(k1 − k2)2(k1 + k2)

2(k1 − k3)2(k1 + k3)

2(k2 − k3)2(k2 + k3)

2 =0

should be satisfied. From this result, we can see that there is a at least a three solitonsolution if the parameter a equals 1, 0 or 1

2. This is a very useful feature of the program

since it allows determination of the values of parameters for which the condition is sat-isfied. In practice, this could used in a search for integrable equations. In the expansionθi = kix− ωit + δi with ω = k3. The coefficient of the two soliton solution is

a12 =(k1 − k2)

2(ak21 − k2

1 − 2ak1k2 − k1k2 + ak22 − k2

2)

(k1 + k2)2(ak21 − k2

1 + 2ak1k2 + k1k2 + ak22 − k2

2).

One could now carry on with either one of the above special values for parameter aand test if there exists a four soliton solution. Or one could construct the three solitonsolutions for the original equation with a = 1, 0 or 1

2.

æ

6. CONCLUSION

In this thesis we developed two symbolic programs, both written in MACSYMA syntax, for theconstruction of soliton solutions. The algorithm of both programs is based on Hirota’s bilinearmethod which allows to construct multiple-soliton solutions of nonlinear evolution and waveequations, provided a bilinear representation of the equations is known. The first programallows to construct one, two and three soliton solutions of a single bilinear equation; the secondone carries out similar computations for a bilinear system. Both programs also allow to test forthe existence of up to four soliton solutions. The programs have been tested on many equationsfrom soliton theory, including well-known single equations and systems.

It is conjectured that every nonlinear PDE that admits a four soliton solution is completelyintegrable. In that sense the symbolic computation of soliton solutions plays a significant rolein testing the integrability of nonlinear PDEs. Presently, there are no other programs availableto test for the existence of exact soliton solutions or to construct these solutions explicitly.

The computer implementation of Hirota’s bilinear method, is made considerably easier byintroducing an equivalent but simpler expression for Hirota’s bilinear operators Dn

x and Dmx Dn

t .To avoid ‘exponential’ expression swell during the calculations, the exponentials are abbreviatedby functions h(x, t) which have inherently all the needed properties of exponentials.

Since the computations are symbolic, bilinear operators can have undetermined parame-ters. The symbolic programs allows to determine the values of these parameters for which theequations will admit soliton solutions, and thus be completely integrable.

It is possible to develop new symbolic programs that calculate soliton solutions of a largefamily of single and/or coupled bilinear equations. With these programs in place, one couldstart a comprehensive search for nonlinear evolution and wave equations that have solitonsolutions. The design of such a program and the investigation of general bilinear forms (usingthis software) would be an interesting topic for further research.

æ

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[25] Nozaki, K. Hirota’s Method and the Singular Manifold Expansion. In: J. Phys. Soc. Jpn.56 (1987): 3052-54.

[26] Newell, A. C., Tabor, M., and Zeng, Y. B. A Unified Approach to Painleve Expansions.In: Physica 29D (1987): 1-68.

[27] Steeb, W. H., and Euler, N. Nonlinear Evolution Equations and Painleve Test. Singapore:World Scientific, 1988.

[28] Gibbon, J. D., Radmore, P., Tabor, M. and Wood, D. The Painleve Test and Hirota’sMethod. In: Stud. in Appl. Math. 72 (1985): 39-63.

[29] Rosales, R. Exact Solitons of Some Nonlinear Evolution Equations. In: Stud. in Appl.Math. 59 (1978): 117-151.

[30] Symbolics, Inc. MACSYMA Reference Manual, Version 13. Cambridge: Symbolics, 1988.

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APPENDIX

A. PROOFS

In this Appendix we give the explicit proofs of formulae (3.12), (4.1) and (4.2) in the text.

A.1 The Proof of Hirota’s N-Soliton Condition

For the bilinear representation of a PDE,

P (Dx, Dt)(f ·f) = 0, (A.1)

the polynomial P is supposed to be even P (Dx, Dt) = P (−Dx,−Dt) and without a constantterm P (0, 0) = 0.

The general N -soliton solution is

f =∑

µ=0,1

exp

(N)∑i<j

Aijµiµj +N∑

i=1

µiθi

, (A.2)

whereθi = kix− ωit + δi , i = 1, 2, . . . , N, (A.3)

and where the constants ki, ωi must satisfy the dispersion relation

P (ki,−ωi) = 0 , i = 1, 2, . . . , N. (A.4)

Furthermore,

aij = exp Aij = −P (ki − kj,−ωi + ωj)

P (ki + kj,−ωi − ωj), i < j . (A.5)

For a three soliton solution to exist, one requires that

S[P, n] =∑

σ=±1

P

(n∑

i=1

σiki,−n∑

i=1

σiωi

)

×(n)∏i<j

(σiki − σjkj,−σiωi + σjωj)σiσj = 0, (A.6)

for each n = 2, . . . , N . Where∑

σ=±1 indicates the summation over all possible combinations

of σ1 = ±1, σ2 = ±1, . . . , σn = ±1 and∏(n)

i<j means the product over all possible combinationsof n elements under the condition i < j.

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Proof: Substituting (A.2) into (A.1) and using (2.7) yields

∑µ=0,1

∑µ′=0,1

P

(N∑

i=1

(µi − µ′i)ki,−N∑

i=1

(µi − µ′i)ωi

)

× exp

N∑i=1

(µi + µ′i)θi +(N)∑i<j

(µiµj − µ′iµ′j)Aij

= 0, (A.7)

where P is given by (A.1). Let the coefficient of the factor

exp

n∑i=1

θi + 2m∑

i=n+1

θi

(A.8)

on the left-hand side of (A.7) be G. It follows that

G =∑

µ=0,1

∑µ′=0,1

cond(µ, µ′)P

(N∑

i=1

(µi − µ′i)ki,−N∑

i=1

(µi − µ′i)ωi

)

× exp

(N)∑i<j

(µiµj − µ′iµ′j)Aij

, (A.9)

where the notation cond(µ, µ′) means that summations over µ and µ′ are taken under thefollowing conditions:

µj + µ′j = 1 for j = 1, 2, . . . , n ,

µj = µ′j = 1 for j = n + 1, n + 2, . . . ,m ,

µj = µ′j = 0 for j = m + 1, m + 2, . . . , N . (A.10)

Defining the variableσj = µj − µ′j , (A.11)

using (A.10) and setting Aij = 0 for i ≥ j, we then have

(N)∑i<j

(µiµj + µ′iµ′j)Aij

=N∑

i=1

N∑j=1

(µiµj + µ′iµ′j)Aij

=

n∑i=1

n∑j=1

+n∑

i=1

m∑j=n+1

+n∑

i=1

N∑j=m+1

+m∑

i=n+1

n∑j=1

+m∑

i=n+1

m∑j=n+1

+N∑

i=n+1

N∑j=m+1

+N∑

i=m+1

n∑j=1

+N∑

i=m+1

m∑j=n+1

+N∑

i=m+1

N∑j=m+1

(µiµj + µ′iµ′j)Aij

T-4162 37

=

n∑i=1

n∑j=1

+m∑

i=n+1

n∑j=1

+m∑

i=n+1

m∑j=n+1

(µiµj + µ′iµ′j)Aij

=n∑

i,j=1

(µiµj + µ′iµ′j)Aij +

m∑i=n+1

n∑j=1

Aij +m∑

i,j=n+1

Aij

=n∑

i,j=1

[1

4(1 + σi)(1 + σj) +

1

4(1− σi)(1− σj)

]Aij

+m∑

i=n+1

n∑j=1

Aij +m∑

i,j=n+1

Aij

=n∑

i,j=1

1

2(1 + σiσj)Aij +

m∑i=n+1

n∑j=1

Aij +m∑

i,j=n+1

Aij

=(n)∑i<j

1

2(1 + σiσj)Aij +

m∑i=n+1

n∑j=1

Aij +m∑

i,j=n+1

Aij . (A.12)

Since σi and σj take the values +1 or −1 for 1 ≤ i, j ≤ n, regarding (A.10) and (A.11), weobtain, from the relations P (k,−ω) = P (−k, ω) and as a consequence of (A.5),

exp[1

2(1 + σiσj)Aij

]= −P (σiki − σjkj,−σiωi + σjωj)

P (ki + kj,−ωi − ωj)σiσj. (A.13)

Substituting (A.11), (A.12) and (A.13) into (A.9) yields

G = c∑

σ=±1

P

(n∑

i=1

σiki,−n∑

i=1

σiωi

) (n)∏i<j

P (σiki − σjkj,−σiωi + σjωj)σiσj , (A.14)

where c is a constant that is independent of the summation indices σ1, σ2, . . . , σN . If we canverify the identity

∑σ=±1

P

(n∑

i=1

σiki,−n∑

i=1

σiωi

) (n)∏i<j

P (σiki − σjkj,−σiωi + σjωj)σiσj = 0 (A.15)

for n = 2, . . . , N , then (A.2) is an exact solution of the bilinear equation (A.1). The aboveidentity is nothing else then S[P, n] = 0; this completes the proof.

A.2 The Proof of the New Expressions for Hirota’s Operators

Hirota’s operators Dnx(f ·g) and Dm

x Dnt (f ·g) given in (2.2) and (2.3) can be written as

Dnx(f ·g) =

n∑j=0

(−1)(n−j)n!

j!(n− j)!

∂jf

∂xj

∂n−jg

∂xn−j, (A.16)

Dmx Dn

t (f ·g) =m∑

j=0

n∑i=0

(−1)(m+n−j−i)m!

j!(m− j)!

n!

i!(n− i)!

∂i+jf

∂ti∂xj

∂n+m−i−jg

∂tn−i∂xm−j. (A.17)

These expressions were used in the algorithms of the two programs.

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Proof: We will prove (A.16) by mathematical induction. The formula (A.17) can be provenin a similar way.

We first try to show that(∂

∂x− ∂

∂x′

)n

(f(x)·g(x′)) =n∑

j=0

(−1)(n−j)n!

j!(n− j)!

∂jf(x)

∂xj

∂n−jg(x′)

∂x′ n−j. (A.18)

For n = 1, (∂

∂x− ∂

∂x′

)(f(x)·g(x′)) =

∂f(x)

∂x·g(x′)− f(x)·∂g(x′)

∂x′

=1∑

j=0

(−1)(1−j)1!

j!(1− j)!

∂jf(x)

∂xj

∂1−jg(x′)

∂x′ 1−j, (A.19)

(A.18) obviously holds. If we assume (A.18) to be true for n− 1, then(∂

∂x− ∂

∂x′

)n−1

(f(x)·g(x′)) =n−1∑j=0

(−1)(n−1−j)(n− 1)!

j!(n− 1− j)!

∂jf(x)

∂xj

∂n−1−jg(x′)

∂x′ n−1−j. (A.20)

Using (A.20), we have(∂

∂x− ∂

∂x′

)n

(f(x)·g(x′)) =

(∂

∂x− ∂

∂x′

)(∂

∂x− ∂

∂x′

)n−1

(f(x)·g(x′))

=

(∂

∂x− ∂

∂x′

)n−1∑j=0

(−1)(n−1−j)(n− 1)!

j!(n− 1− j)!

∂jf(x)

∂xj

∂n−1−jg(x′)

∂x′ n−1−j

=n−1∑j=0

(−1)(n−1−j)(n− 1)!

j!(n− 1− j)!

(∂j+1f(x)

∂xj+1

∂n−1−jg(x′)

∂x′ n−1−j

− ∂jf(x)

∂xj

∂n−jg(x′)

∂x′ n−j

)

=n−1∑j=0

(−1)(n−1−j)(n− 1)!

j!(n− 1− j)!

∂j+1f(x)

∂xj+1

∂n−1−jg(x′)

∂x′ n−1−j

−n−1∑j=0

(−1)(n−1−j)(n− 1)!

j!(n− 1− j)!

∂jf(x)

∂xj

∂n−jg(x′)

∂x′ n−j

=n∑

j=1

(−1)(n−j)(n− 1)!

(j − 1)!(n− j)!

∂jf(x)

∂xj

∂n−jg(x′)

∂x′ n−j

+n−1∑j=0

(−1)(n−j)(n− 1)!

j!(n− 1− j)!

∂jf(x)

∂xj

∂n−jg(x′)

∂x′ n−j

=n−1∑j=1

((−1)(n−j)(n− 1)!

(j − 1)!(n− j)!+

(−1)(n−j)(n− 1)!

j!(n− 1− j)!

)∂jf(x)

∂xj

∂n−jg(x′)

∂x′ n−j

+∂nf(x)

∂xn+ (−1)nf(x)

∂ng(x′)

∂x′ n

T-4162 39

=n−1∑j=1

(−1)(n−j)n!

j!(n− j)!

∂jf(x)

∂xj

∂n−jg(x′)

∂x′ n−j

+∂nf(x)

∂xn+ (−1)nf(x)

∂ng(x′)

∂x′ n

=n∑

j=0

(−1)(n−j)n!

j!(n− j)!

∂jf(x)

∂xj

∂n−jg(x′)

∂x′ n−j.

Thus, (A.18) is true for all n = 1, 2, . . . . Setting x′ = x, we have

Dnx(f ·g) =

n∑j=0

(−1)(n−j)n!

j!(n− j)!

∂jf(x)

∂xj

∂n−jg(x)

∂xn−j.

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B. FLOWCHART OF HIROTA SINGLE.MAX

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T-4162 41

Fig. B.1: Flowchart of HIROTA SINGLE.MAX

C. TABLE WITH LOGICAL COMBINATIONS

In this Appendix we give the table with the logical combinations for the Boolean variables thatcan be selected in the batch files.

æ

Tab. C.1: Logical Combinations of batch files

D. PROGRAM CODES AND TEST OUTPUT

In this Appendix we give code of the programs HIROTA SINGLE.MAX and HIROTA SYSTEM.MAXand some MACSYMA output of the test cases.

D.1 HIROTA SINGLE.MAX

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D.2 HIROTA SYSTEM.MAX

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D.3 Output of Test Cases

D.3.1 Korteweg-de-Vries (KdV) Equation

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D.3.2 Kadomtsev-Petviashvili (KP) Equation

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T-4162 57

.

D.3.3 Modified Korteweg-de-Vries (mKdV) Equation

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D.3.4 Higher Order Ito Coupled Bilinear Equations

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D.3.5 Bilinear Equations with a Parameter

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