Analytical travelling wave solutions and parameter analysis for the (2+1)-dimensional Davey–Stewartson-type equations

PRAMANA c© Indian Academy of Sciences Vol. 81, No. 5— journal of November 2013

physics pp. 747–762

Analytical travelling wave solutions and parameter analysisfor the (2+1)-dimensional Davey–Stewartson-typeequations

JIANPING SHI1,∗, JIBIN LI2 and SHUMIN LI1

1Center for Nonlinear Science Studies, Kunming University of Science and Technology,Kunming, Yunnan 650093, People’s Republic of China2Department of Mathematics, Zhejiang Normal University, Jinhua,Zhejiang 321004, People’s Republic of China∗Corresponding author. E-mail: [email protected]

MS received 6 April 2013; revised 22 July 2013; accepted 26 July 2013DOI: 10.1007/s12043-013-0612-6; ePublication: 22 October 2013

Abstract. By using dynamical system method, this paper considers the (2+1)-dimensionalDavey–Stewartson-type equations. The analytical parametric representations of solitary wave solu-tions, periodic wave solutions as well as unbounded wave solutions are obtained under differentparameter conditions. A few diagrams corresponding to certain solutions illustrate some dynamicalproperties of the equations.

Keywords. Periodic wave solution; solitary wave solution; unbounded wave solution; Davey–Stewartson-type equations; parameter analysis.

PACS Nos 05.45.Yv; 03.65.Ge; 47.35.Fg

1. Introduction

The nonlinear evolution equations play an important role in applied mathematics andphysics. Many researchers tried their best to find analytical solutions or numerical solu-tions to the nonlinear evolution equation by using various methods, because the solutionsdescribed the phenomena on physics and other fields. The Davey–Stewartson equation(DS equation) was derived by Davey and Stewartson in 1974 [1]. Much work had beendone in the past few decades to improve this equation. Especially, in 1988, Boiti et al[2] considered the DS equation and constructed solutions that decay exponentially in alldirections. Using inverse scattering method, Fokas and Santini [3,4] derived such solu-tions and they named these objects as dromion. Hietarinta and Hirota [5,6] and Radhaand Lakshmanan [7] gained multidromion solutions or one-dromion solutions to DSequation by bilinear method. Their works developed the DS equation in an interesting

Pramana – J. Phys., Vol. 81, No. 5, November 2013 747

Jianping Shi, Jibin Li and Shumin Li

way. Recently, many new methods such as sine–cosine method [8], double exp-functionmethod [9], so called G ′/G method [10], etc. which were used to construct exact solu-tions of DS equation had been established and developed. The solutions of DS equationhave been applied in plasma physics, nonlinear optics, hydrodynamics, etc. For exam-ple, the solutions of DS equation could describe the interaction between a spatiotemporaloptical plus and adequately matched microwaves [11].

In this paper, we consider the following (2+1)-dimensional Davey–Stewartson-typeequations (n-DS for short) with power law nonlinearity (see [10,12] and cited referencestherein):

iqt + a(qxx + qyy) + b|q|2nq = αqr,

rxx + ryy + β(|q|2n)xx = 0, (1.1)

where q(x, y, t) is a complex valued function which denotes the complex amplitude,while r(x, y, t) is a real valued function which denotes the velocity of an underlyingmean flow. a, b, α, β are real parameters and n (>0) is the power-law parameter. Wealways assume that a �= 0. A pair of coupled equations is used to describe the long-timeevolution of a two-dimensional wave packet in the original physical context. The authorsof [10] and [12] obtained some exact explicit solutions of (1.1). To our knowledge, thedynamical behaviour of the travelling wave system (1.1) has never been studied before.In this paper, the dynamical system method is used to investigate system (1.1) in its para-metric space. Many possible analytical explicit solutions of (1.1) can be obtained. Thedynamical properties of solutions are analysed by the diagrams of the travelling wavesolutions. The study of this paper gives rise to some new results for eq. (1.1).

Similar to Ebadi and Biswas [10], we study the solutions of (1.1) having the form

q(x, y, t) = u(ξ) exp(iθ), r(x, y, t) = v(ξ), (1.2)

where

ξ = a1x + a2 y + a3t, θ = θ1x + θ2 y + θ3t.

In the sequel, we denote q(k)n-DS and r (k)

n-DS as the kth-type solution of system (1.1) of theform (1.2).

Substituting (1.2) into (1.1) and integrating the obtained equations twice with ξ , wehave the parameter relationship a3 = −2a(θ1a1 + θ2a2) and

v(ξ) = − βa21

a21 + a2

2

u2n (1.3)

−(θ3 + aθ21 + aθ2

2 )u + a(a21 + a2

2)u′′ +

(b + αβa2

1

a21 + a2

2

)u2n+1 = 0, (1.4)

where the integral constant is assumed to be zero and ′ denotes the derivative about ξ .This paper is organized as follows. In §2, we consider the exact solutions of (1.1)

when n = 1 and n = 2 (1-DS and 2-DS for short). In §3, for an arbitrary real number

748 Pramana – J. Phys., Vol. 81, No. 5, November 2013

(2+1)-dimensional Davey–Stewartson-type equations

n > 0, n �= 1, 2, solutions of the general form of (1.1) (n-DS for short) are studied.Properties of three chosen solutions are illustrated by some diagrams drawing in theappropriate parameter conditions in §4. A comprehensive conclusion is given in §5.

2. Analytic solutions of 1-DS and 2-DS equations

Let us first investigate all solutions of (1.1) of the form (1.2) when n = 1, 2. In thiscondition, eq. (1.4) is equivalent to the following two Hamiltonian systems, respectively:

du

dξ= y,

dy

dξ= δu − γ u3 (2.1)

and

du

dξ= y,

dy

dξ= δu − γ u5, (2.2)

where

δ = (θ3 + aθ21 + aθ2

2 )

a(a21 + a2

2), γ =

(b + [αβa2

1/(a21 + a2

2)])

a(a21 + a2

2).

Equations (2.1) and (2.2) have the following Hamiltonian integrals, respectively:

H1(u, y) = 1

2y2 − 1

2δu2 + 1

4γ u4 (2.3)

and

H2(u, y) = 1

2y2 − 1

2δu2 + 1

6γ u6. (2.4)

For δγ > 0, systems (2.1) and (2.2) have three equilibrium points O(0, 0) andE1,2(±ue, 0) where ue = ue1 = (δ/γ )1/2 and ue = ue2 = (δ/γ )1/4, respectively.For δγ < 0, systems (2.1) and (2.2) have only one equilibrium point O(0, 0). In thiscase, we have ho = H1,2(0, 0) = 0, he1 = H1(ue1, 0) = −δ2/4γ , he2 = H2(ue2, 0) =−(δ/3)(δ/γ )1/2.

It is easy to obtain the bifurcations of phase portraits of (2.1) and (2.2) which are shownin figures 1a–1d.

2.1 Analytical solutions of 1-DS equation

We consider three cases based on different signs of δ and γ .

Case 1. Suppose that δ > 0, γ > 0 (see figure 1a). In this case, system (2.1) is theDuffing oscillator.



(a) > 0, γ > 0.δ (b) < 0, γ > 0.δ

(c) < 0, γ < 0.δ (d) > 0, γ < 0.δFigure 1. The bifurcations of phase portraits of systems (2.1) and (2.2) in the (δ, γ )-plane.

Corresponding to the two families of periodic orbits defined by H1(u, y) = h, h ∈(he1, 0), the formula

y2 = γ

2(a2

1(h) − u2)(u2 − b21(h))

can be obtained from (2.3), where

a21(h) = 1

γ(δ +

√δ2 + 4γ h), b2

1(h) = 1

γ(δ −

√δ2 + 4γ h).

Thus, we have the following parametric representations of two families of periodic orbits(see [13]):

u(ξ) = ±a1(h)dn(ωξ, k), (2.5)



where

ω =√

γ

2a1(h), k =

√a2

1(h) − b21(h)

a21(h)

.

Therefore, eq. (2.5) gives rise to two families of periodic wave solutions of (1.1) asfollows:

q(1)1-DS = ± exp(iθ)a1(h)dn(ωξ, k),

r (1)1-DS = − βa2

1

a21 + a2

2

(a1(h)dn(ωξ, k))2. (2.6)

When h → 0, k → 1, a21(h) → 2δ/γ. Corresponding to the two homoclinic orbits with

‘figure eight’ of (2.1) defined by H1(u, y) = 0, we have the parametric representation ofu(ξ) as follows:

u(ξ) = ±√

2δ

γsech(

√δξ). (2.7)

It implies the following analytical solutions of (1.1):

q(2)1-DS = ±

√2δ

γexp(iθ) sech(

√δξ),

r (2)1-DS = − βa2

1

a21 + a2

2

(√2δ

γsech(

√δξ)

)2

. (2.8)

Corresponding to the family of periodic orbits defined by H1(φ, y) = h, h ∈ (0,∞),the formula

y2 = γ

2(a2

1(h) − u2)(u2 + (−b21(h))), b2

1(h) < 0

can be obtained from (2.3). Thus, we have the following parametric representation of thefamily of periodic orbits:

u(ξ) = a1(h)cn(ω1ξ, k1), (2.9)

where k1 = 1/k, k is given by (2.5), ω1 =√

(γ (a21(h) − b2

1(h)))/2.

Thus, (2.9) gives rise to a family of periodic wave solutions of (1.1) as follows:

q(3)1-DS = ± exp(iθ)a1(h)cn(ω1ξ, k1),

r (3)1-DS = − βa2

1

a21 + a2

2

(a1(h)cn(ω1ξ, k1))2. (2.10)

Case 2. Suppose that γ > 0, δ ≤ 0 (see figure 1b).

Corresponding to the family of periodic orbits defined by H1(u, y) = h, h ∈ (0,∞), wehave the same parametric representation of u(ξ) as (2.9). So, we have the same solutionsas (2.10).



Case 3. Suppose that δ < 0, γ < 0 (see figure 1c).

Corresponding to the family of periodic orbits defined by H1(u, y) = h, h ∈ (0, he1), wehave

y2 = |γ |2

(a22(h) − u2)(b2

2(h) − u2)

from (2.3), where

a22(h) = 1

|γ | (|δ| +√

δ2 + 4γ h), b21(h) = 1

|γ | (|δ| −√

δ2 + 4γ h).

Thus, we have the following parametric representation of the family of periodic orbits:

u(ξ) = b2(h)sn(ω3ξ, k2), (2.11)

where

k2 = a2(h)√a2

2(h) + b22(h)

, ω3 =√ |γ |

2a2(h).

It means that the analytical solutions of (1.1) is

q(4)1-DS = ± exp(iθ)b2(h)sn(ω3ξ, k2),

r (4)1-DS = − βa2

1

a21 + a2

2

(b2(h)sn(ω3ξ, k2))2. (2.12)

The two heteroclinic orbits of (2.1) defined by H1(φ, y) = he1 have the parametricrepresentations

u(ξ) = ±(√

δ

γ

)tanh

(√ |δ|2

ξ

). (2.13)

Hence we have the following analytical solutions of (1.1):

q(5)1-DS = ±

√δ

γexp(iθ) tanh

(√ |δ|2

ξ

),

r (5)1-DS = − βa2

1δ

(a21 + a2

2)γtanh2

(√ |δ|2

ξ

). (2.14)

2.2 Analytical solutions of 2-DS equation

In this case, we only need to consider the orbits of (2.2) in the right phase plane. Supposethat δ > 0, γ > 0 (see figure 1a).

Corresponding to the right family of periodic orbits defined by H2(u, y) = h, h ∈(he2, 0), we have

y2 = γ

3

(6h

γ+ 3δ

γu2 − u6

)= γ

3

(6h

γ+ 3δ

γz − z3

)

= γ

3(r1 − z)(z − r2)(z − r3), z = u2



from (2.4). By using the first equation of (2.2), we obtain

γ

3ξ =

∫ z

r2

dz√(r1 − z)(z − r2)(z − r3)

. (2.15)

Therefore, (2.15) gives rise to

u(ξ) = √z(ξ) =

(r2 + (r1 − r2)(r2 − r3)sn2(1ξ, k)

(r1 − r3) − (r1 − r2)sn2(1ξ, k)

)1/2

, (2.16)

where

1 = γ√

(r1 − r3)

6, k =

√(r1 − r2)

(r1 − r3).

It gives rise to a family of periodic wave solutions of (1.1) as follows:

q(1)2-DS = exp(iθ)

(r2 + (r1 − r2)(r2 − r3)sn2(1ξ, k)

(r1 − r3) − (r1 − r2)sn2(1ξ, k)

)1/2

,

r (1)2-DS = − βa2

1

a21 + a2

2

(r2 + (r1 − r2)(r2 − r3)sn2(1ξ, k)

(r1 − r3) − (r1 − r2)sn2(1ξ, k)

)2

. (2.17)

When h → 0, k → 1, r1 → √3δ/γ , r2 → 0, r3 → −√

3δ/γ . Corresponding tothe right homoclinic orbit of (2.2) defined by H2(u, y) = 0, we have the parametricrepresentation of u(ξ) as follows:

u(ξ) =(

3δ

γ

)1/4

(sech(δξ))1/2. (2.18)

It implies the following analytical solution of (1.1):

q(2)2-DS =

(3δ

γ

)1/4

exp(iθ)sech1/2(δξ), r (2)2-DS = − 3δβa2

1

(a21 + a2

2)γsech2(δξ).

(2.19)

3. Analytical solutions of n-DS equation in the case n > 0, n �= 1, 2

For any real number n > 0, n �= 1, 2, making the transformation u(ξ) = φ1/n(ξ), then(1.4) becomes

− (θ3 + aθ21 + aθ2

2 )n2φ2 + a(a21 + a2

2)(1 − n)(φ′)2 + a(a21 + a2

2)nφφ′′

+(

b + αβa21

a21 + a2

2

)n2φ4 = 0. (3.1)



Equation (3.1) is equivalent to the system

dφ

dξ= y,

dy

dξ= α1 y2 + nφ2(δ − γφ2)

φ, (3.2)

where α1 = 1 − (1/n), δ and γ are same as above.This system has the first integral

H(φ, y) = φ−2α1

(y2 + nδ

α1 − 1φ2 − nγ

α1 − 2φ4

)= h. (3.3)

In refs [14–17], the second author of this paper and his partners had studied the systemwhich has the analogous form of eq. (3.2). They also named this system as the first classof singular travelling wave equations with the singular straight line φ = 0. There aredifferent dynamical behaviours between the solutions of (3.2) and a smooth dynamicalsystem. Furthermore, the following equation can be obtained from (3.3):

y2 = hφ2α1 − nδ

α1 − 1φ2 + nγ

α1 − 2φ4 ≡ hφ2α1 + F(φ), (3.4)

where

F(φ) = −φ2

(nδ

α1 − 1− nγ

α1 − 2φ2

).

As α1 �= 1, 2, for all h �= 0, it is very difficult to compute the exact parametric repre-sentations of all orbits corresponding to the first equation of (3.2). However, if h = 0, theexact parametric representations of the orbits of (3.2) can be achieved.

The geometric properties of all solutions of (3.2) are shown by phase portraits for everygiven n > 0 and a fixed parameter pair (δ, γ ). Thus, we can gain the analytical travellingwave solutions of (1.1) in the form of u(x, t) = u(ξ) = (φ(ξ))1/n . In order to get realsolutions, we only consider the case of φ(ξ) ≥ 0 whether n is odd or even.

By using the method proposed in [14,15], we study the associated regular system

dφ

dζ= yφ,

dy

dζ= α1 y2 + nφ2(δ − γφ2), (3.5)

where dζ = φ dξ. System (3.5) has the same phase orbits as (3.2). Just as Li and Dai [15]pointed out, near the straight line φ = 0, there are ξ = εζ, ε 1, which means that ξ isa variable with ‘slow time-scale’ compared to the time-scale of the variable ζ .

For δγ > 0, system (3.5) has three equilibrium points O(0, 0) and E1,2(±φe, 0) whereφe = (δ/γ )1/2. For δγ < 0, system (3.5) has only one equilibrium point O(0, 0).

M(φe, 0) is denoted as the coefficient matrix of the linearized system of (3.5) at theequilibrium point (φe, 0) and J (φe, 0) = det M(φe, 0). We have

J (0, 0) = 0, J (φe, 0) = 2nδ2

γ.

Thus, we can confirm the type of equilibrium points of system (3.5) by the value of J (see[14]). (0, 0) is a two-order equilibrium point, and if γ > 0, then J > 0, (φe, 0) is a centrepoint, and if γ < 0, then J < 0, (φe, 0) is a saddle point.



At these points, we have

ho = H(0, 0) = 0, he = H(φe, 0) = − n3δ

n + 1

(δ

γ

)1/n

.

By qualitative analysis, we obtain the bifurcations of phase portraits of (3.5) shown infigures 2a–2d.

By using the phase portraits shown in figure 2, we discuss the dynamical behaviour ofthe solutions of (3.2) as follows:

Case 1. Suppose that δ > 0, γ > 0 (see figure 2a).

(a) > 0, γ > 0.δ (b) < 0, γ > 0.δ

(c) < 0, γ < 0.δ (d) > 0, γ < 0.δFigure 2. The bifurcations of phase portraits of system (3.5) in the (δ, γ )-plane.



Corresponding to H(φ, y) = h, h ∈ (he, 0), there are two families of closed orbits of(3.2) which give rise to two families of periodic travelling wave solutions of (1.1).

Corresponding to H(φ, y) = 0, there are two homoclinic orbits of (3.2) (see figure 3a).We see from (3.4) that

y2 = n2γ

n + 1φ2

((n + 1)δ

γ− φ2

).

Thus, using the first equation of (3.2) to calculate, we have the following exact parametricrepresentations:

φ(ξ) = ±√

(n + 1)δ

γsech

(n√

δξ)

. (3.6)

(a) > 0, γ > 0.δ (b) < 0, γ < 0.δ

(c) > 0, γ < 0.δFigure 3. The level curves defined by H(φ, y) = 0 for system (3.5).



It gives rise to the analytical solution of (1.1) as follows:

q(1)n-DS = exp(iθ)

(√(n + 1)δ

γsech

(n√

δξ))1/n

,

r (1)n-DS = − βa2

1

a21 + a2

2

(√(n + 1)δ

γsech

(n√

δξ))2

. (3.7)

Corresponding to H(φ, y) = h, h ∈ (0,∞), this is a family of closed orbits of (3.2)(see figure 2a), which contact the singular straight line φ = 0. It implies that the closedfamily brings about a family of periodic travelling wave solutions of (1.1) (see [14–16]).

Case 2. Suppose that δ < 0, γ > 0 (see figure 2b).

In this case, corresponding to H(φ, y) = h, h ∈ (0,∞), there are two families of closedorbits of (3.2), which contact the singular straight line φ = 0. It is obvious that theseclosed families bring about one or two families (for n is an odd number) of periodictravelling wave solutions of (1.1) (see [14–16]).

Case 3. Suppose that δ < 0, γ < 0 (see figure 2c).

Corresponding to H(φ, y) = h, h ∈ (0, he), there are two families of closed orbitsof (3.2), which contact the singular straight line φ = 0. It is obvious that these closedfamilies bring about one or two families (for n is an odd number) of periodic travellingwave solutions of (1.1) (see [14–16]).

Corresponding to H(φ, y) = he, there exist two homoclinic orbits of (3.2), whichcontact the singular straight line φ = 0. It implies that the system consists of one or two(for n is an odd number) solitary travelling wave solutions of (1.1) (see [14–16]).

For h = 0, the level set defined by H(φ, y) = 0 are two open curves (see figure 3b).We see from (3.4) that

y2 = n2|γ |n + 1

φ2

(φ2 − (n + 1)δ

γ

).

Thus, using the first equation of (3.2), the following analytical parametric representationscan be obtained:

φ(ξ) = ±√

(n + 1)δ

γsec

(n√|δ|ξ

). (3.8)

Hence, we have

q(2)n-DS = exp(iθ)

(√(n + 1)δ

γsec

(n√|δ|ξ

))1/n

,

r (2)n-DS = − βa2

1

a21 + a2

2

(√(n + 1)δ

γsec

(n√|δ|ξ

))2

. (3.9)

This parametric representation defines an unbounded solution of (1.1).



Case 4. Suppose that δ > 0, γ < 0 (see figure 2d).

For h = 0, the level set defined by H(φ, y) = 0 are four open curves consisting of thestable and unstable manifolds of the origin O(0, 0) (see figure 3c). We see from (3.4) that

y2 = n2|γ |n + 1

φ2

(φ2 + (n + 1)δ

|γ |)

.

Thus, using the first equation of (3.2), the following analytical parametric representationscan be obtained:

φ(ξ) = ±√

(n + 1)δ

|γ | csch(

n√

δξ − 0

), (3.10)

where

0 = arcsh

(√[((n + 1)δ)/|γ |]φ0

), φ(0) = φ0 > 0.

Equation (3.10) implies the analytical parametric representations of the solution of (1.1)as follows:

q(3)n-DS = exp(iθ)

(√(n + 1)δ

|γ | csch(

n√

δξ − 0

))1/n

,

r (3)n-DS = − βa2

1

a21 + a2

2

(√(n + 1)δ

|γ | csch(

n√

δξ − 0

))2

. (3.11)

4. The influence of the parameters on solutions

In this section, we consider some properties of 1-DS equation by simulating theparametric representations in different parameter conditions.

4.1 Periodic solutions

We consider solutions (2.6) as the case of periodic solution. In the parameter condition(a = b = α = β = 1, a1 = a2 = 1, θ1 = θ2 = θ3 = 1), given t = 1, figure 4a showsthe real part, imaginary part of q(1)

1-DS and r (1)1-DS. It is obvious that amplitude of the wave

changes when change happens independent of parameters a, b, α or β (see figure 4b).When a1 and a2 are equal, the wave keeps its shape, but if they are inconsistent, the wavecan change its shape immediately (see figure 4c), the parameters θ1 and θ2 have the samecharacters (see figure 4d).

4.2 Solitary wave solutions

We consider solutions (2.8) as the case of solitary wave solutions corresponding to thetwo homoclinic orbits. In order to have a full view of the solution, we suppose y = 1,



(a) a = b = = 1, a1 = a2 = 1 = 2 = 1.=α β θ θ

(b) a = 3, b = = 1, a1 = a2 = 1 = 2 = 1.=α β θ θ

(d) a = b = = 1, a1 = a2 = 1, 1 = 1.2, 2 = 0.3=α β θ θ

(c) a = 0.8, b = 3, = 1, a1 = 0.8, a2 = 1.3, 1 = 2 = 1.=α β θ θ

Figure 4. The diagrams defined by the solutions of the form (2.6).

and we can see the real part, imaginary part of q(2)1-DS and r (2)

1-DS in figure 5. It is obviousthat the position of the wave shifts when the parameter a is changed (see figure 5b). Theparameter θ1 also influences the amplitude and shape of the waves (see figures 5c and 5d).

4.3 Solutions of the form (2.14)

The solutions (2.14) correspond to the heteroclinic orbits. It means kink charactersaccording to Li and Dai [15]. Figure 6 illustrates the real part, imaginary part of q(5)

1-DS,r (5)

1-DS and the parametric representations (2.13). It is obvious that the solutions displaydifferent forms under different parameter conditions.



(a) a = b = = 1, a1 = a2 = 1 = 2 = 1.=α β θ θ

(b) a = 0.5, b = = 1, a1 = a2 = 1 = 2 = 1.=α β θ θ

(d) a = b = = 1, 1 = 2, a1 = a2 = 2 = 1.=α β θ θ

(c) a = b = = 1, 1 = 0.5, 2 = 1.=α β θ θa1 = a2 =

Figure 5. The diagrams defined by the solutions of the form (2.8).



(a) a = –1, b = 1, = 0.8,α = 0.3.β

(b) a = –2, b = 1, = 0.8,α = 0.3.β

(c) a = –1, b = 3, = 0.8,α = 0.3.β

(d) a = –2, b = 3, = 0.8,α = 0.3.βFigure 6. The diagrams defined by the solutions of the form (2.14).

5. Conclusion and remarks

To sum up, we have the following main results from the above discussions:

Theorem 5.1. Considering the (2+1)-dimensional Davey–Stewartson-type eq. (1.1) withthe form (1.2), depending on the parameter groups (a, b, α, β) and (a1, a2, a3, θ1, θ2, θ3),eq. (1.1) is transformed into new systems with respect to parameter group (δ, γ )

according to the dynamical system approach. Then, there are two results as follows:



(1) When n = 1, 2, eq. (1.1) is transformed into Hamiltonian systems (2.1) and (2.2),under different parameter conditions given in §2, eq. (1.1) has seven analyticalsolutions defined by (2.6), (2.8), (2.10), (2.12), (2.14), (2.17), (2.19).

(2) When n > 0, n �= 1, 2, eq. (1.1) is transformed into the associated regular system(3.5) which has the first integral (3.3). Considering H(φ, y) = 0, under differentparameter conditions given in §3, eq. (1.1) has three analytical solutions defined by(3.7), (3.9), (3.11).

Remark 1. Notice that if n > 0, n �= 1, 2, h �= 0, it is very difficult to compute the exactparametric representations because the regular system (3.5) has the first integral (3.3)including a special part φ−2α1 . We have to find another method to study this question.

Remark 2. The simulations in §4 show that if one or two parameters change, the wavecorresponding to the solution can change its amplitude, shape or other characters. It meansthat the exact analytical solution may be helpful for studying the physical properties ofthe problem associated with eq. (1.1).

Acknowledgements

The authors thank the referees for their valuable suggestions. This work was supportedby the National Natural Science Foundation of China (10831003).

References

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Analytical travelling wave solutions and parameter analysis for the (2+1)-dimensional Davey–Stewartson-type equations