Nonlinear Evolution Equations and Inverse Scattering

By

Abd-Alrahman Mahmoud Shehada Jabr

Supervisor

Prof.Dr. Gharib Mousa Gharib

This Thesis was Submitted in Partial Fulfillment of the Requirements for

the Master’s Degree of Science in Mathematics

Faculty of Graduate Studies

Zarqa University

Zarqa - Jordan

November , 2016

ii

جامعة الزرقاء

نموذج التفويض

المؤسسات للمكتبات أوأنا عبدالرحمن محمود شحادة جبر، أفوض جامعة الزرقاء بتزويد نسخ من رسالتي

األشخاص عند طلبهم حسب التعليمات النافذة في الجامعة. أو الهيئات أو

التوقيع:

التاريخ:

Zarqa University

Authorization Form

I, Abd-Alrahman Mahmoud Shehadah Jabr, authorize Zarqa University to supply

copies of my Thesis to libraries or establishments or individuals on request,

according to the University regulations.

Signature:

Date:

iii

COMMITTEE DECISION

This Thesis (Nonlinear Evolution Equations and Inverse Scattering) was

successfully defended and approved on ______________

Examination Committee Signature

Dr. Gharib Mousa Gharib (Supervisor)

Prof. of Mathematics

Dr. Khaled Khalil Jaber (Member)

Assoc. Prof. of Mathematics

Dr. Naser Hassan Al-Zomot (Member)

Assoc. Prof. of Mathematics

Dr. Ali Mahmud Ateiwi (Member)

Prof. of Mathematics

iv

ACKNOWLEDGEMENT

I am very grateful to many people for the completion of this project, the first of them

and above all the Almighty Allah for the countless blessing, after that I especially thank my

supervisor, Head of the Department of Mathematics, Prof.Dr. Gharib Musa Gharib for his skilled

guidance and unwavering support throughout this project. I am most grateful to the Department of

Mathematics at Zarqa University and all faculty members.

I also extend my thanks to my family in general and specially my mother and father for

their Doa’a without which, this thesis could not have been accomplished. Finally I would like to

thank my wife for her support, cooperation and patience.

v

Table of Contents Authorization Form ii

Committee Decision iii

Acknowledgement iv

Table of Contents vi

Abstract vii

Introduction 1

Chapter One: Background

1.1 Partial differential equation ……………………………………….………..………(3)

1.2 Discover of IST …………………………………………………………….………(4)

1.3 Soliton solution……………………………………………………...………….…..(6)

1.4 Nonlinear evolution equation…...…………………………………..……………....(7)

Chapter Two: Inverse Scattering Transform

2.1 The components of the IST method……………………………………….....(9)

2.2 Linear Example of the Inverse Scattering Transform……………………..……..…(10)

2.3 The IST for the KdV equation……..…………………………………………….…(14)

2.4 One soliton solution for thKdV equation………….…...………………………….. (23)

2.5 two soliton solution for thKdV equation……………………...…………………... (27)

vi

Chapter Three: The Family of Equations

3.1 The AKNS system and IST for the family of equation……………………...……....... (34)

3.2 Example1 The Sine-Gordon equation…………………………………………..…...…(40)

3.3 Example1 The Sinh-Gordon equation………………..…………………..…………….(42)

3.4. Conclusions…………..…………….…………………………………..…..….………...(44)

3.5. References………………………………………………………………..……..……….(45)

Abstract(in Arabic)……………...……….………………………………………….………..(48)

vii

ABSTRACT

The main original contribution of this thesis is the development of the inverse scattering

transform (IST) method for nonlinear evolution equations. The equation we solve a general form

of KdV equation, which is known as fully integrable model. In chapter one, we introduce

historical background about PDE and some kind of solutions for PDE, also we talk about a

nonlinear evolution equation which is very important in so many phenomena of waves, solitary

waves, and soliton solution. In chapter 2, we introduce IST method for solving these equations

and we write classifications for the integrable models of equations which are solvable by our main

method IST and solve a linear example and summarize this method for solving KdV equation. In

chapter 3, we talk about the family of equations and introduce the AKNS system for this family

and give some examples for the family of equations like Sine-Gordon equation, Sinh-Gordon

equation and Liovell’s equation.

1

INTRODUCTION

Differential equations are important because they explain relationships containing rate of

change. Such relationships form the basis for studying phenomena in the sciences, engineering,

economics , medicine and most human knowledge .

Since the study of differential equations began, almost, problem had been solved only for

linear equations “ those have at most one power of the unknown variables or its derivatives

appeared in each term of the equation”. The main reason for this is that linear equations can be

solved by the superposition principle . That is, since differentiation is a linear operation, any

linear combination of solutions of a linear equation is again a solution of the equation. Hence,

the methods of Fourier series and Fourier transforms were developed to find the general

solutions in terms of sums or integrals of certain basic solutions.

However, in the last 40 years, the mathematicians focus in the study of nonlinear equations and

in methods for their exact solution. A nonlinear partial differential equation relates to study a

number of different physical systems like water waves, an harmonic lattices, plasma physics

and elastic rods. It describes the long time evolution of amplitude dispersive waves.

The field of nonlinear waves and integrable systems has a long and difficult history, see

[1,2,3,5]. It began in the nineteenth century with the pioneering work of Stokes, Boussinesq

and Korteweg and de Vries [1], all of whom studied the dynamics of fluids. Many of the

models that were derived were nonlinear partial differential equations, and without

computational assistance very little could be said at the time about their solutions. In the

second half of the twentieth century, some of these models were then rediscovered by

researchers such as Kruskal and Zabusky in 1965, they used a combination of mathematical

analysis and computational power to explain the Fermi-Pasta-Ulam (FPU) paradox. There was

an observation of recurring states of energy (rather than the expected dissipation) within a one-

dimensional string of connected masses with nonlinear spring interactions[1-5].

From studies of properties of the equation and its solutions, the concept of solitons was

introduced and the method for exact solution of the initial-value problem using inverse

scattering theory was developed[1]. The recent literature contains many extensions for

2

nonlinear evolution equations of physical interest and to other classes of equations. Some of

these equations and results are introduced. In this thesis , we will give a historical account of

some of these equations, specifically concerning the method of inverse scattering transform

(IST).

In this thesis, we consider the one-dimensional problems solvable by the IST method. The IST

is also connected to Backlound transformations, which relate solutions of partial differential

equations to solutions of other equations. Also more recently were done for higher dimensional

problems. The IST is one of the most important developments in mathematical physics, which

is applied to solve many linear partial differential equations. The name “ inverse scattering

method “ comes from the idea of recovering the time evolution of a potential from the time

evolution of its scattering data [3,5,8].

Now before we study this method to find exact solution for nonlinear evolution equation we

will introduce a classification of partial differential equation .

3

Chapter 1

Background

1.1: Partial differential equation

Recall that the words differential and equations clearly indicate solving some kind of equation

involving derivatives. We have seen the classification of partial differential equations into

linear, quasi linear, semi linear, homogeneous, non-homogeneous and nonlinear[1,4,7,8].

1.1.1 Definition ( partial differential equation ). A partial differential equation (PDE) is an

equation that has an unknown function depending on at least two variables, contains some

partial derivatives of the unknown function [5].

1.1.2 Definition An evolution equation is a partial differential equation for an unknown

function of the form

where is an expression involving only u and its derivatives with respect to . If this

expression is nonlinear, equation (1.1) is called a nonlinear evolution equation (NLEE) [1].

1.1.3 Definition. A solution to PDE is, generally speaking, any function (in the independent

variables) that satisfies the PDE. However, from this family of functions one may be uniquely

selected by imposing adequate initial and/or boundary conditions.

A PDE with initial and boundary conditions constitutes the so-called initial-boundary-value

problem (IBVP). Such problems are mathematical models of most physical phenomena [3,4].

1.1.4 Definition. (A well-posed problem) An initial-boundary-value problem is well-posed if:

it has a unique solution,

and the solution varies continuously with the given inhomogeneous data, that is, small

changes in the data should cause only small changes in the solution [11].

4

1.2: Discover of Inverse Scattering Transform :

In 1965 Zabusky and Kruskal explained the FPU problem in terms of solitary wave solutions to

the Korteweg-de Vries (KdV) equation. In their numerical analysis they observed “solitary-

wave pulses”, these pulses are called “solitons” because of their particle-like behavior, and

noted that such pulses interact with each other nonlinearly but come out of interactions

unaffected in size or shape except for some phase shifts [3]. At that time no one knew how to

solve the IVP for the KdV equation, except numerically. In 1967 Gardner, Greene, Kruskal,

and Miura presented a method, now known as the IST to solve that IVP, assuming that the

initial profile decays to zero sufficiently rapidly as | | . They showed that the

integrable nonlinear partial differential equation (NPDE), i.e. the KdV equation[1-6],

is related to a linear ordinary differential equation (LODE), which is the one-dimensional

Schraodinger equation,

( )

where is the eigenvalue associate the eigenfunction and that the solution to

(1.2) can be recovered from the initial profile . They also explained that the soliton

solutions to the KdV equation correspond to a zero reflection coefficient in the associated

scattering data. Note that the variables x and t in (1.2) are spatial and time variable

respectively and they throughout denote the partial derivatives with respect to those variables

[1,2,3].

In 1972 Zakharov and Shabat showed [20], that the IST method is applicable also to the IVP

for the nonlinear Schraodinger (NLS) equation,

| |

5

where i denotes the imaginary number √ . They proved that the associated linear differential

equation is the first-order linear system,

{

where λ is the spectral parameter and denotes complex conjugation for . The

system (1.5) is now known as the Zakharov-Shabat system [2].

After that, again in 1972 Wadati showed that the IVP for the modified Korteweg-de Vries

(mKdV) equation see [1,3,4],

can be solved by the inverse scattering problem for the linear system,

{

Next, in 1973 Ablowitz, Kaup, Newell, and Segur showed that [1,2,4,5] the IVP for the Sine-

Gordon equation,

can be solved in the same way by the inverse scattering problem associated with the linear

system,

{

Since then, many other NPDEs have been discovered to be solvable by the IST method.

6

1.3: Soliton solution

The first observation of a soliton was made in (1834) by the Scottish engineer John Scott

Russell at the Union Canal between Edinburgh and Glasgow [3,4,5]. Russell reported his

observation to the British Association of the Advancement of Science in September (1844), but

he did not succeed in convincing the scientific community.

The Dutch mathematician Korteweg and de Vries published [14] a paper in (1895) based on de

Vries’ Ph.D. dissertation, in which surface waves in shallow water, narrow canals were

modeled by what is now known as the KdV equation. The importance of this paper was not

understood until (1965), even though it contained as a special solution what is now known as

the one-soliton solution.

1.3.1 Definition. A Soliton is the part of a solution to an integrable nonlinear partial

differential equation due to a pole of the transmission coefficient inS the upper half complex

plane.

The soliton solution that satisfies nonlinear equations usually has the following properties [3]:

1) solitons are waves dying out at infinity and they have profiles, which are unaltered after

colliding with other solitons.

2) they evolve with time and therefore they satisfy certain evolution equations.

3) they are stable solutions and they do not disperse apart when they collide with other

solitons.

4) in collision with other solitons, there is nonlinear interaction.

However, they retain their original shape shortly afterwards, only slightly displased.

5) soliton with larger amplitude pulse moves faster and is narrower in width

than the smaller soliton .

7

1.4: Nonlinear evolution equation.

A nonlinear evolution equation focuses on the study of a number of different physical systems

like water waves, plasma physics, harmonic lattices, and elastic rods.

1.4.1 KdV equation

The equation that Kruskal and Zabusky[1], found a model as

where δ is a parameter, which is a NLEE in two independent variables and is known as the

KdV equation. This is in fact the equation found by Korteweg and de Vries when they studyied

shallow water waves. These waves are localised waves, unlike linear waves, interact elastically

with neighbouring waves, and have a relationship between amplitude and speed,this particle-

like nature led is called solitons.

An example of a two-soliton solution of the KdV equation

.

[ ] /

whose graph as a function of and is shown in Figure 1[1,13].

FIGURE 1a. 3-D Two-soliton of the KdV equation

8

FIGURE 1b. Two-soliton the KdV equation

1.4.2 Nonlinear Schrödinger equation and sine-Gordon equation

After the KdV system was found there was a great number of advancements as researchers

found ways of applying this new method of solution to a number of important systems. One of

the important applications was in (1971) from Zakharov and Shabat [5], who used ideas of Lax

to solve the initial-value problem for the nonlinear Schrödinger equation

|

|

for solutions with decaying boundary conditions. Like the case of the KdV equation, the

researchers found a soliton solutions and an infinite number of conservation laws. In (1972)

Wadati solved the mKdV (2.4), and in (1973) Ablowitz, Kaup, Newell and Segur (AKNS) [1-

5] applied this method to solve the sine-Gordon equation,

for which they found soliton solutions, breather solutions and an infinite number of

conservation laws. The wide applicability of this method then led AKNS [4] [5] to show that

equations (1.10), (1.11), (1.12) and (1.13) are in fact all related to a single matrix eigen value

problem, from which many physically important systems are obtainable. Noting the similarity

between this method of solving partial differential equations and the method of Fourier

transform, they also labelled it the IST.

9

Chapter 2

Inverse Scattering Transform

The Inverse Scattering Transform (IST) is a method of finding solutions to linear and

integrable nonlinear partial differential equations. In this chapter, we look at some definitions

and the mathematical structure of the IST and its application to solve the heat equation and

nonlinear KdV equation.

2.1: The components of the IST method :

2.1.1 Direct scattering problem : The problem of determining the scattering data

corresponding to a given potential in a differential equation.

2.1.2 Inverse scattering problem : The problem of determining the potential that corresponds

to a given set of scattering data in a differential equation.

2.1.3 Lax method : A method introduced by Lax in 1968 that determines the integrable

nonlinear partial differential equation associated with a given linear ordinary differential

equation so that the initial value problem for that nonlinear partial differential equation can be

solved with the help of an inverse scattering transform.

2.1.4 Scattering data : The scattering data associated with a linear ordinary differential

equation usually consists of a reflection coefficient which is a function of the spectral

parameter , a finite number of constants that correspond to the poles of the transmission

coefficient in the upper half of the complex plane, and the bound-state norming constants,

whose number for each bound-state pole is the same as the order of that pole. It is desirable

that the potential in the linear ordinary differential equation is uniquely determined by the

corresponding scattering data and vice versa.

10

2.1.5 Time evolution of the scattering data : The evolvement of the scattering data from its

initial value at to its value at a later time .

In this section we well introduced an important PDE and applied the IST to find exact solution

of them.

The Heat equation,

and the KdV equation,

The heat equation and the KdV equation are both partial differential equations in one spatial

dimension which is and one temporal dimension which is , however one fundamental

difference between these two equations is that (2.1) is linear in , but (2.2) is not. Despite this

difference, they satisfy the compatibility condition for an associated linear system, which is the

basis for the IST. The IST for the heat equation is simple, because of the fact that the equation

itself is linear.

2.2: Solution of the heat equation by Inverse Scattering Transform

Consider the heat equation (2.1) with an initial condition satisfying

∫| |

this can be solved by separation of variables or Fourier transform to give the general solution

as,

∫

11

where is the Fourier transform of the initial condition . Alternatively one can

generate [5] the Lax pair, which is a pair of matrices or operators satisfy Lax equation

whose consistency gives . Equation (2.5a) arising the forward scattering

problem, we solve it for along the initial time . The second Lax equation (2.5b)

is then used to evaluate from which may then be constructed from (2.5a).

To solve (2.5a) for we start with the boundary condition| | , but satisfies

Faddeev condition then | | , then we have

then the solution of is

| |

for some constant . We then introduce the unique Jost solutions and 𝜑 of (2.5a) with the

boundary conditions,

|𝜑 |

| |

in terms of the initial condition we have,

𝜑 ∫

∫

by (2.3) these integrals will exist in the half-planes and

12

respectively. Furthermore since these are both solutions of (2.5a), it follows that

𝜑

for some function B which is independent of , and we can write B as,

∫

That is the direct scattering procedure. These expressions are not enough to determine the time

dependent Jost solutions 𝜑 and since this would involve the knowledge of

, which is the same thing needed to find. Now consider the time dependence of the

function , defined by,

𝜑

Substitute this equation into equation (2.5b) and taking the limit we have

since that and as for all , then we have

𝜑

∫

∫

13

and so provided that is bounded and that

∫| |

it follows that

𝜑 (

) | |

(

) | |

So we have

𝜑

∫ (

)

Thus the solution of the heat equation becomes

∫ (

)

∫

This is clearly the result where we recognize as the Fourier transform of the initial

condition

2.3: Inverse Scattering Transform for the KdV equation

The steps for solving the KdV equation by the IST are more complicated than the heat

equation. This method was first discovered by Gardner, Greene, Kruskal and Miura [1-5] in the

14

1960s. Here we give an important mathematical features of these steps, and for a detailed

analysis on the forward scattering problem considered here see [3], while the inverse problem

is treated in [6]. All mathematicians agree with that the Lax pair for the KdV equation is given

by

where is constant depends on the normalization of and is the spectrum parameter. The

direct problem at time , given . The spectrum of these equations consists of a

finite number of discrete eigenvalues, , , for and a continuum,

, for < 0 [1,2,3].

Suppose that , then the Lax pair for KdV equation becomes

where, provided that since and are independent of time then, , since the

consistency of the system of to satisfy the KdV equation, then

first equation (2.22a) defines the forward scattering problem, while the second (2.22b) defines

the time evolution of the scattering problem. Assume that there exists some real-valued initial

condition which satisfies

∫| | | |

15

which is associate to integrability condition, this initial condition is called Faddeev condition

[3].

The forward scattering problem is used to determine to

This equation is a second-order equation which is linear for , and it is called a Sturm-

Louiville equation. Now to solve this equation we first consider the limit

| | , in which equation (2.22a) becomes

and the solution for becomes

| |

for some constant A and B. Then we introduce two unique Jost solutions by the boundary

conditions

2𝜑

��

2

since equation (2.27) is invariant under the transformation , it follows that 𝜑

�� and . Furthermore since the solution of equation (2.27) is

two-dimensional space we may write

𝜑

��

16

where the two functions , and are independent of , which

satisfy

where , are the complex conjugate of respectively[1,3].

More of that the functions and satisfy the condition

and this satisfies

| | | |

Now consider the Wronskian equation

𝜑 �� 𝜑�� 𝜑 ��

on . Note that this implies that A does not vanish on . We now state

several properties about the Jost solutions and the functions A and B.

Proposition 2.3.1. The Jost solutions and the spectral functions A and B have the following

analyticity properties:

𝜑 and exist and they are continuous at the eigenvalue in the closed

half-plane Im( ) ≥ 0, and are analytic in in the open half-plane .

�� and exist and they are continuous at the eigenvalue in the closed

half-plane , and are analytic in in the open half-plane .

exists and is continuous in in the closed half-plane , and is analytic in

in the open half-plane .

B(0; ) exists and is continuous in on .

17

To prove that see [1,3,4,5].

Proposition 2.3.2. The Jost solutions and the spectral functions A and B have the following

asymptotic properties:

{𝜑 (

)

(

)

| |

{�� (

)

(

)

| |

(

) | |

(

) | |

Theorem 2.3.3.The function has a finite number of zeroes in the open half-

plane , and does not vanish on . Moreover all of these zeroes are simple

and lie on the imaginary axis . At each we have 𝜑 =

for some constant to prove see [1] .

Now, we study the evolution of the normalization constant for eigenfunction 𝜑 By

definition, the normalization constant is defined as :

{ ∫ 𝜑

}

18

In order to study the evolution of the normalization constant, we differentiate both sides with

respect to the time and use the second Lax equation 𝜑 𝜑 𝜑

∫ 𝜑𝜑

∫ 𝜑

𝜑 𝜑 𝜑

∫

𝜑 𝜑 𝜑 𝜑

∫ 𝜑

𝜑 𝜑 𝜑 𝜑 𝜑

∫ 𝜑

[ 𝜑 𝜑 𝜑 𝜑 ]

the solution for the above equation gives us the equation of evolution for the normalizing

constant

After that the transmission and reflection coefficients since λ gives rise to unbound state ,

then defines the following

𝜑

19

0𝜑 𝜑 𝜑

𝜑1

(

𝜑) 0

( )

1

whence, in order to eliminate these exponential functions, we must equate the coefficients of

and . Simply, we may first rewrite the above expression as

0

1

0

1

since and are linearly independent then

in order to vanish the second term, the coefficient must be equal to zero

similarly

𝜑

20

[𝜑 𝜑 𝜑

]

0

1

and the relationship between the reflection and transmission coefficient satisfies,

| | | |

Finally, we can state the following which summarizes our analysis for the scattering data.

If are given as above then

{

where and are obtained from the initial data for the

KdV equation

The n zeroes of A form a set of discrete eigenvalues, which we will show are in fact associated

with the N solitons which exist within the solution of the KdV equation.

The inverse transform involves using A and B to reconstruct the time dependent solution

As for the heat equation this is done from the setting of a Riemann-Hilbert problem.

Importantly, since the boundary conditions for u are independent of time, the analyticity and

asymptotic results of Propositions 2.3.1 and 2.3.2 and Theorem 2.3.3 continue to hold for all

. By these results the relation

21

𝜑

.

/

defines a condition between the two functions

and along the real -axis, with known

boundary condition. For Im( ) > 0, the solution of this is given by the singular integral

∑.

/

∫ .

/

This is a closed-form singular integral equation for , where all time dependence is

known from .

One can also isolate the dependence of on the spectral parameter by taking the form

∫

now to find K, inserting expression (2.53) into (2.35), thereby obtaining a Goursat problem for

K. It can be shown that the solution of this Goursat problem exists and is unique [7]. The

motivation for this option comes from the fact that one of the boundary conditions for K gives

a simple relation between it and the solution to the KdV equation:

[ ]

In order to obtain an equation for we substitute equation (2.53) into the singular

integral equation for . Since K is related to the Fourier transform of , by taking the inverse

Fourier transform we obtain the following Gel’fand-Levitan equation, valid for :

22

∫

where the quantity L is given by

∑.

/

∫ .

/

furthermore by considering the Wronskian one can also show that

( ∫

)

which follows from the fact that the Jost solutions are real whenever . Thus from

the knowledge of

{ { }}

one can construct the full time-dependent solution through the linear Volterra-type

integral equation (2.52). The quantity

is known as the transmission coefficient and

is known as the reflection coefficient. Initial conditions for which the reflection

coefficient is identically zero on are known as reflectionless potentials.

Now, we study two examples of soliton solution.

23

2.4: One soliton solution

Consider the initial value problem

With the initial condition , where n is the number of soliton, let

, then we have

where corresponds to the vertical displacement of the water from the equilibrium at

the location at time . Replacing by amounts to replacing by in (2.58). Also, by

scaling and , i.e. by multiplying them with some positive constants, it is possible to

change the constants in front of each of the three terms on the left-hand side of (2.58).

Solution

First we find the eigenvalues and eigenfunction

( )

where

Let this transformation map from for to [-1,1] for s.

So we have

Therefore, the Sturm-Louiville problem becomes

24

[

] [

]

Comparing this equation with the associated Legender equation

[

] 0

1

we get

| |

It is clear that this is the only one eigenvalue and the corresponding eigenfunction can be found

from the Lgender polynomials,

⁄

where n=0,1,2,…,N, and

let N=1 then we have,

⁄

⁄

⁄

25

So the eigenvalue and the eigenfunction

Second we Normalize the eigenfunction

∫ | |

∫ | |

Therefore, the normalized eigenfunction is

√

Third determination of and

by using the definition

√

√

Therefore, the evolution equation for the normalization constant is given by

√

After that, we determine integration kernel

∑

∫ (

)

∑

26

Then we write Gel’fand and Levitan equation

∫

∫

We solve the above equation by the separation of variables by assuming

Substituting (2.72) into (2.71), we get

∫

Comparing coefficients of in equation (2.73) gives

∫

∫

So we have

Multiply by

, so we get

so the kernel becomes

27

and the potential can be solved as

.

/

(

)

Thus the solution becomes

FIGURE 1. One-soliton solutions of the KdV equation.

2.5: Two soliton solution

Consider the initial value problem

28

with the initial condition let

Solution

First we find the eigenvalues and eigenfunctions

( )

where

Let this transformation map from for x to [-1,1] for s.

So we have

Therefore, the Sturm-Louiville problem becomes

[

] [

]

Comparing this equation with the generalized Legender equation

[

] 0

1

we get

| |

These are two eigenvalues for the Sturm-Louiville problem and in order to find the

corresponding eigenfunctions, we use the associated Legender polynomials with ,

29

⁄

⁄

.

/

⁄ ⁄

and

(

)

So the eigenvalues and the eigenfunctions

After that, normalization of the eigenfunction

∫ | |

∫ | |

∫ | |

∫ | |

Therefore, the normalized eigenfunctions are

√

√

Then, we determine and

30

by using the definition

√

√

√

√

Therefore, the evolution equation is given by

√

√

After that, determination of integration kernel

∑

∫ (

)

∑

31

Then we write Gel’fand and Levitan equation

∫

∫ [ ]

We solve the above equation by separation of variables by assuming

Then by substitution in the above we have

∫ [ ][

]

Comparing coefficients of and equation (2.93) gives

∫ [

]

[ ] [ ]

The second equation is

∫ [ ][

]

∫

∫

32

[ ] [ ]

From ( 2.95a ) and ( 2.96b ), we have the following system:

{[ ] [

]

[ ] [ ]

The above system of algebraic equations can be solved by using Cramer’s rule

where

|

|

|

|

|

|

Substituting the above results into equation (2.92), we obtain

[ ] [ ]

Thus

[ ]

To derive the solution

33

. [ ]

/

(

)

where

Thus the solution becomes

FIGURE 3. Two-soliton solutions of the KdV equation.

34

Chapter 3

The Family of Equations

In the previous chapter, we introduced a historical and steps of the IST, as we saw the IST is a

very difficult method and has a lot of scattering data, so in this chapter we will introduce this

method to solve a general formula of important -dimention of equations which describes

a pseudospherical surfaces (pss) and gives some examples and compare the solution of these

equations with the solutions given by other methods.

First we define that equation which describes pss, that is an equation describes a surface with

Gaussian curvature .

Beals, Rabelo and Tenenblat (BRT) [18] introduced the family of equations as follows:

[ ]

where

where is a differentiable function of with are real constants, such

that This family includes the sine-Gordon, sinh-Gordon and Liouville’

equations.

3.1 The AKNS system and IST for the family of equation :

Let be two-dimensional differentiable function space with coordinates A DE for a

real function describes a pss. If it’s a necessary and sufficient condition for the

existence of differentiable functions [3]

Dependent of and its derivatives such that the form

35

satisfies the definition of equations that describes pss i.e

The above definition is equivalent to that DE for is the integrability condition for the

problem so we can write

(𝜑

)

where d denotes exterior differentiation, Ƥ is 2Χ2 matrix such that

(

)

Take

(

)

where which is a parameter independent of and while and are functions of

and , now we have [3]

If we assume that the above equations are compatible, that satisfies that , then P and

Q must satisfy

[ ]

36

and

consider the 2 x 2 scattering problem

{𝜑 𝜑

𝜑

and the linear time dependence is given by

{𝜑 𝜑 𝜑

where A, B, C and D are scalar functions of and λ, independent 𝜑 . Now

, we just specify that

(

) (

)

Now, when , then (3.8) reduced to the Schrodinger scattering problem

It is interesting to note that the most interesting nonlinear evolution equations phenomena

when or (or if q is real). This procedure provides a simple technique

which allows us to find a nonlinear evolution equations expressible in the form (3.7). The

compatibility condition of equations (3.8-9), that is requiring that 𝜑 𝜑 , and

assuming that the parameter is time-independent, that is , allows a set of conditions

which A, B, C and D must satisfy. Therefore

𝜑 𝜑

37

𝜑 𝜑

𝜑 𝜑 𝜑

𝜑 𝜑 𝜑

Respectively, do the same thing to then we have the following results

So without loss of generality we assume that then we have

Since the solution is related to take

and . and if we choose A, B and C

as

.

/

(

)

(

)

Then A,B and C satisfy equation (3.16), so the linear time dependence is

{

𝜑

.

/𝜑

(

)

(

)𝜑

.

/

38

Since we have

where

(

)

The solution of equation (3.19) is

where is constant column vector i.e

.

/

(

)

Now we choose ( ) then we have

{𝜑

[ (

) ]

[ (

) ]

Konno and Wadati introduced the function

𝜑

This function first appeared and explained the geometric context of pss equations, now

39

Now we write the kernel integral as

∑

∫

writing Gel’fand and Levitan equation

∫

∑

2 0 .

/ 13

. 0 .

/ 1/

Thus the solution of the family of equation

2 0 .

/ 13

40

3.2: Example 1: the sine-Gordon equation :

In the family of equation if we choose and

then we have

where is real valued and the subscripts denote the partial derivatives with respect to the

spatial coordinate x and the temporal coordinate t. This equation is called the sine-Gordon

equation, The solution of the sine-Gordon equation depends on the solution of the family of

equation as follows.

So we start with the scattering problem

{𝜑 𝜑

𝜑

If we choose and

called the potential corresponding to a solution of

the sine-Gordon equation .

So we have the following lax pair

{𝜑 𝜑

𝜑

We notice that, if 𝜑 are bounded and sufficiently rapidly as , then

2𝜑

And we have

41

(

)

(

)

(

)

So the linear time dependence is

{𝜑 (

)𝜑 (

)

(

)𝜑 (

)

Since the solution is related to through

so that

∫

it follows that , as . Then we have

{𝜑

𝜑

And the solution becomes

, * (

) +-

42

3.3: Example 2: the Sinh-Gordon equation :

In the family of equation if we choose and

then we have

where is real valued and the subscripts denote the partial derivatives with respect to the

spatial coordinate x and the temporal coordinate t. This equation is called the Sinh-Gordon

equation. The solution of the Sinh-Gordon equation depends on the solution of the family of

equation.

So we start with the scattering problem

{𝜑 𝜑

𝜑

If we choose and

called the potential corresponding to a solution of

the sine-Gordon equation .

So we have the following lax pair

{𝜑 𝜑

𝜑

We notice that, if 𝜑 are bounded and sufficiently rapidly as , then

2𝜑

And we have:

43

(

)

(

)

(

)

So the linear time dependence is

{𝜑 (

)𝜑 (

)

(

)𝜑 (

)

Since the solution is related to through

so it follows that , as .

{𝜑

𝜑

And the solution becomes

{ [ (

) ]}

44

3.4: Conclusion

The application of the inverse scattering transform (IST) as a method of solving

physically relevant nonlinear partial differential equations has much grown and

used since its discovery in the late 1960s. It provides a way of linearizing these

systems, that is a means of obtaining their solutions through the solving of linear

equations. Some of the major successes of the IST in mathematical physics have

been the solving of the Korteweg-de Vries equation and its variants, the nonlinear

Schrodinger equation and the sine- Gordon equation, all of which have great

importance in the field. The IST gives a wide class of solutions satisfying

decaying boundary conditions, which are typically the most physically relevant

scenarios.

We found in chapter 2 that inverse scattering transform is similar to

Fourier transform but Fourier transform cannot solve the nonlinear models

because we have unsolvable integral equation when we do the inverse Fourier

transform unlike inverse scattering transform. After that we concluded that the

new inverse scattering method with using Tanh-transform to solve the linear

system of Lax pair gave identical results to the results of the original way and

we found that in one soliton and two soliton solution examples. And we can see

that when we use this transform making the IST easier.

After that in chapter 3 we solved the family of equation which included

the Sine-Gordon and Sinh-Gordon equations, then we saw that the solution of

these equations by the general solution of the family of equation is related to the

solution of these equations in several method of solving that kind of equations.

45

3.5: References

[1] Ablowitz, M.J. and P.A. Clarkson, 1991. Solitons, Nonlinear Evolution

Equations and Inverse Scattering Transform. Cambridge Univ. Press, Cambridge.

[2] M. Ablowitz, G. Biondini, and B. Prinari. Inverse Scattering Transform

for the Integrable Discrete Nonlinear Schrodinger Equation with Nonvanishing

Boundary Conditions. Inv. Prob., 23(1711-1758), 2007.

[3] M. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform, SIAM,

Philadelphia, 1981.

[4] M. Ablowitz, D. Kaup, A. Newell, and H. Segur. Method for Solving

the Sine-Gordon Equation. Phys. Rev. Lett., 30:1262–1264, 1973.

[5] M. Ablowitz, D. Kaup, A. Newell, and H. Segur. The Inverse Scattering

Transform - Fourier Analysis for Nonlinear Problems. Stud. Appl. Math.,

53:249–315, 1974.

[6] M. Ablowitz and J. Ladik. On the Solutions of a Class of Nonlinear

Partial Difference Equations. Stud. Appl. Math., 57:1–12, 1977.

[7] B. Dubrovin, V. Matveev, and S. Novikov. Nonlinear Equations of

Korteweg-de Vries Type, Finite Zoned Linear Operators, and Abelian Varieties.

Russ. Math. Surv., 31:59–146, 1976.

[8] B. Dubrovin and S. Novikov. Periodic and Conditionally Periodic Analogues

of the Many-soliton Solutions of the Korteweg-de Vries Equation.

Sov. Phys. JETP, 40:1058–1063, 1974.

46

[9] Gardner, Clifford S.; Greene, John M.; Kruskal, Martin D.; Miura, Robert M.

(1967), "Method for Solving the Korteweg-deVries Equation", Physical Review

Letters 19: 1095–1097,

[10] Gardner, Clifford S.; Greene, John M.; Kruskal, Martin D.; Miura, Robert

M. (1974), "Korteweg-deVries Equation and Generalization. VI. Methods for

Exact Solution.", Comm. Pure Appl. Math. 27: 97–133,

doi:10.1002/cpa.3160270108, MR 0336122

[11] H. Blohm. Solutions of a Discrete Inverse Scattering Problem and the

Cauchy Problem of a Class of Discrete Evolution Equations. J. Math.

Phys., 40:4374–4392, 1999.

[12] J. Shaw, Mathematical Principles of Optical Fiber Communications, SIAM,

Philadelphia, 2004.

[13] J. Atkinson, J. Hietarinta, and F. Nijhoff. Seed and Soliton Solutions of

Adler’s Lattice Equation. J. Phys. A, 40:F1–F8, 2007.

[14] Liu, S., Z. Fu, S. Liu and Q. Zhao, 2001. Jacobi Elliptic Function Expansion

Method and Periodic Wave Solutions of Nonlinear Wave Equations. Physics

Letters A, 289: 69-74.

[15] M. Boiti, F. Pempinelli, B. Prinari, and A. Spire. An Integrable

Discretization of KdV at Large Times. Inv. Prob., 17:515–526, 2001.

47

[16] N. Asano, Y. Kato, Algebraic and Spectral Methods for Nonlinear Wave

Equations, Longman Scientific & Technical, Essex, England, 1990.

[17] R.K. Bullough, P.J. Caudrey. "Solitons" Topics in Current Physics 17.

Springer Verlag, Berlin-Heidelberg-New York, 1980.

[18] Rogers, C. and W.F. Shadwick, 1982. Backlund Transformations. Aca.

Press, New York.

[19] V. A. Marchenko, "Sturm-Liouville Operators and Applications",

Birkhäuser, Basel, 1986.

[20] V. Adler, A. Bobenko, and Y. Suris. Discrete Nonlinear Hyperbolic

Equations.

Classification of Integrable Cases. Funct. Anal. Appl., 43(1):3–17,

2009.

[21] Zhang, J., D. Zhang and D. Chen, 2011. Exact Solutions to a Mixed Toda

Lattice Hierarchy through the Inverse Scattering Transform. Journal of Physics

A: Mathematical and Theoritical, doi: 10.1088/1751-8113/44/11/115201.

48

ورة ومعكوس التشتتالمعادالت غير الخطية المتط

عدادإ

عبذانشح يحد شحادة جبش

شرافإ

د. غريب موسى غريب

الملخص

انذف انشئض نهشصانت اصخعشاض غشمت يعكس انخشخج نحم انعادالث غش انخطت

انخطسة ي أيثهخا ظاو يعادالث ك د ف ، حث حطشلا ف انجزء األل نهخعشفاث انخعهمت

بانعادالث انخفاظهت حصفا ركش بعط انطشق نحم ز انعادالث خصصا بانزكش انعادالث

ش انخطت انخطسة يذ أخا ف انعهو انفزائت انظاش انطبعت بعذ رنك حعشفا عه غ

بعط أاع انحهل نز انعادالث يثم انحم انج نهعادالث انخفاظهت انحم انضهخ . ثى ف

هعادالث عاللخا انجزء انثا دسصا غشمت يعكس انخشخج حعشفا عه أخا ف إجاد انحم ن

بطشق اخش ، لا بخحهم بعط انخطاث ف ز انطشمت كا غبما ز انطشمت عه يثال خط

ي انعادالث يعادنت حذفك انحشاسة انشسة غبما اعا ز انطشمت عه حانت خاصت ي

خعشظا بعط األيثهت يعادالث ك د ف ثى أدخها ححم خخصش بعط انخطاث ف انحم ثى اص

عه أاع انحم انضهخ نعادنت ك د ف. ثى انجزء انثانث حطشلا نهذف انشئض حم

صسة عايت ي انعادالث حض بعائهت انعادالث حث حشخم عه بعط انعادالث انت يثم

ى خطة إجاد انظاو جسد يعادالث نفم. عذ حم عائهت انعادالث كاج أ-يعادالث كال

انخط انشحبػ بانعائهت يا ض بظاو الكش ثى بعذ رنك أجذا انصففت انعشفت نزا انظاو

ي ثى حابعا خطاث انحم ف غشمت يعكس انخشخج اصخكها عاصش 2 2ف فعاء انعادالث

ه انعكس انخشخج انشاد حه ع انخشخج األياي لا بعم انخطش نز انعاصش نحصم ع

غشك يعادنت حكايهت حض بـ جم فاذ ي ثى إجاد انحم انضهخ بعذ حم انعادنت اصخعشظا حم

جسد ، بماست زا انحم انضهخ جذا -نهعادالث انشخمت ي ز انعادنت يثم يعادالث كال

عادالث .أ ياثم نحهل اخش بطشق اخش نز ان