TRAVELING WAVE SOLUTIONS AND THEIR STABILITY OF NONLINEAR SCHR¨ODINGER EQUATION WITH WEAK DISSIPATION??

Yancong Xu,Tianzhu Lan,Yongli Liu(Dept.of Math.,Hangzhou Normal University,Zhejiang 310036,PR China)

TRAVELING WAVE SOLUTIONS AND THEIR STABILITY OF NONLINEAR SCHR¨ODINGER EQUATION WITH WEAK DISSIPATION??

Yancong Xu?,Tianzhu Lan,Yongli Liu
(Dept.of Math.,Hangzhou Normal University,Zhejiang 310036,PR China)

In this paper,several new constant-amplitude and variable-amplitude wave solutions (namely,traveling wave solutions) of a generalized nonlinear Schr¨odinger equation are investigated by using the extended homogeneous balance method,where the balance method is applied to solve the Riccati equation and the reduced nonlinear ordinary differential equation,respectively.In addition, stability analysis of those solutions are also conducted by regular phase plane technique.

nonlinear Schr¨odinger equation;extended homogeneous balance method;amplitude wave solutions;stability

2000 Mathematics Subject Classification 35B40;35K58;35B32

1　Introduction

The investigation of temporal or spatial dynamics for nonlinear Schr¨odinger equations is an important and interesting subject,see,for example,[1-11,13-18,20] for details.In particular,there are many papers which have paid more attention to the dynamics of the following (2+1) -dimensional cubic nonlinear Schr¨odinger (NLS) equation without dissipation

where u=u (x,y,t) is a complex-valued function,α,β,and γ are real constants,and subscripts represent partial derivatives.As we know,the NLS equation is referred to as an approximate model of the evolution of a nearly monochromatic wave of small amplitude of pulse propagation in Langmuir waves in a plasma,optical fibers and gravity waves on deep water with different values of parameters.

Recently,modulational instability of many extended versions of the NLS equation with different dissipations have been investigated in[14,15,17].In particular, there is a nonlinear dissipative Schr¨odinger (DissNLS) equation as follows:

where u=u (x,y,t) (x,y∈R) is a complex-valued function,α,β,γ,a,b,c and d are real constants with a,b and c being all nonnegative,d represents dissipation.Note that,this equation is regarded as a model of weakly nonlinear surface wave,and it can also be regarded as a generalized version of the complex Ginzburg-Landau equation.

Actually,modulational instability corresponds to temporal stability.However, the investigation of traveling wave solutions also plays an important role in the dynamics of nonlinear physical phenomena,see,for example,Zhang et al.[16],Feng and Meng[19],Nguyen[21].To my best,except for particular parameters,there are no exact analytical solutions of (2+1) -dimensional NLS equation,so sometimes one has to resort to computer numerical simulations in order to investigate the dynamics of NLS,thus it is necessary to obtain exact solutions by certain analytic technique.Therefore,in order to better understand the dynamical behavior of the dissipative nonlinear Schr¨odinger equation (1.2) ,in this paper,we will focus on its exact traveling wave solutions and their spatial stability.

The rest of this paper is outlined as follows.Section 2 contains two kinds of exact amplitude traveling wave solutions obtained by the homogeneous balance method. In Section 3,we study the stability of traveling wave solutions of NLS equation by using the regular phase plane method.

Now,we introduce the homogenous balance method and use it to look for special exact solutions of some nonlinear equations.Consider a general partial differential equation

where H is a polynomial function of its arguments,subscripts denote the partial derivatives.We will solve (1.3) by the homogeneous balance method with the following four steps:

Step 1Firstly,take

where m and n are nonnegative integers,the functions f=f (φ) and φ=φ (x,t) , and the coefficients Aijare all to be determined.Substituting (1.4) into (1.3) ,the integers m and n will be determined.

Step 2Secondly,substituting the linear combination chosen in Step 1 into (1.3) , collecting all terms with the highest order derivatives of φ and setting its coefficient to be zero,we obtain an ordinary differential equation to get the function f.

Step 3Thirdly,substituting the linear combination chosen in Step 2 into (1.3) , the nonlinear terms of various derivatives of f can be replaced by the corresponding higher order derivatives of f.Then collecting all terms with the same order derivatives of f and setting the coefficient of each order derivatives of f to be zero, respectively,we obtain a set of equations for φ and the coefficients of the linear combination in Step 1.If the equations are solved,then φ and the coefficients of the combination can be determined.

Step 4Finally,substituting f,φ,m,n and some constants obtained in Steps 2 and 3 into (1.4) ,we obtain exact solutions of (1.3) .

2　Amplitude Wave Solutions in Terms of Explicit Functions

Now we consider the traveling wave solutions in the variable ξ=kx+ly?vt of the form

where k∈R,l∈R,ω and v represent the frequency and translation speed,respectively,ρ (ξ) and ? (ξ) are real functions of the pseudo-time ξ.After substituting (2.1) into (1.2) and setting the real part and imaginary part to be zero,respectively,we have

where the subscript ξ denotes

Take ?ξ=ψ,then (2.2) can be restated as

Remark 1 Note that,the function ρ (ξ) is a solution of (2.3) ,then?ρ (ξ) is also a solution of (2.3) .So in what follows,we just consider ρ (ξ) ＞0 satisfying (2.3) .

Then we solve (2.3) by the above homogeneous balance method.Let

where φ=φ (ξ) ,f=f (φ) and g=g (φ) are real functions,Ai,Bi,A,B are real constants to be determined,m and n are positive integers to be determined.

We will solve the functions ρ＞0 and ψ by three steps.

Step 1To obtain m and n in (2.4) .

By substituting (2.4) into (2.3) and balancing the highest-order derivative term and the leading nonlinear term in (2.3) ,one yields

In the following,we shall obtain ρ＞0,ψ and ?.

Step 2To solve functions f and g in (2.5) .

Based on (2.5) ,it is easy to get the equalities as follows:

Substituting the above equations into (2.3) and setting the coefficient of the highest power in φξequal to be zero,we have ordinary differential equations about f and g as follows:

where b1,b2are constants to be given.

Now we compute constants b1,b2.Substituting (2.8) into (2.7) ,we obtain the following system of algebraic equations with respect to b1and b2,

(i) When b1=0,b2is an arbitrary constant;

then it is easy to obtain the following solutions of (2.9) :

Actually,in view of (γS+cQ)2≥0, (2.10b) will not be satisfied,here it can not be considered.

Step 3To solve functions ρ (ξ) and ? (ξ) .

We turn to find ρ (ξ) and ψ (ξ) with two cases:b1=0 and b1/=0.

Case Ab1=0.From (2.8) ,we know

Substituting the above two equations into (2.6) ,one obtains

Substituting the above equations into (2.3) ,collecting all terms with the same order derivative of g,that is,g,g′and g′′,and setting whose coefficients to be zero respectively,we have

We can easily obtain two solutions of (2.13) as follows:

where v,ω satisfying

Therefore from (2.5) and (2.9) ,we have

where ξ0is an arbitrary constant,A,B are given by (2.14) and (2.16) ,respectively.

In addition,from (2.5) and (2.8) ,we have

Substituting the above equations into (2.6) ,it follows that

Substituting the above equations into (2.3) ,collecting all terms with the same order derivative of f,namely,f′,f′′and f′′,and setting whose coefficients to be zero respectively and noticing (2.19) ,we know that the following equalities hold,

Now we solve the above system as well as the functions ρ (ξ) and ψ (ξ) in three cases.

We can obtain a solution of (2.21) as follows:

where ω,v,d satisfies the following relations:

Substituting (2.22) into (2.5) ,it is easy to find that

Remark 2 By letting ξ2=0,ξ1=1 in (2.23) ,it follows that

Then we can obtain a solution of (2.25) as follows:

where v,ω satisfies (2.15) and (2.17) ,respectively.

Substituting (2.26) into (2.5) ,it is easy to find that

Now,we can obtain two types of traveling wave solutions in terms of explicit functions:

Type 1 Constant-amplitude traveling wave solutions.

From (2.14) , (2.16) and (2.18) ,in view of the symmetry u→?u,we obtain the following constant-amplitude solutions

Remark 3Note that,in L¨u et al.[12],u1and u2are actually the same solutions,as well as u3and u4,u5and u6by noting that if u is a solution of (1.2) , then?u is also a solution of (1.2) .

Type 2 Variable-amplitude traveling wave solutions.

From (2.23) and (2.27) ,we obtain two variable-amplitude traveling wave solutions,

where A,λ and B are given by (2.22) ,see Fig.1.

where A,λ and B are given by (2.26a) and (2.26b) .ξ0,ξ1and ξ2are arbitrary constants withare given by (2.10a) .Note that,in view of different parameter values,there should be eight kinds of solutions.For simplicity, here we only consider four kinds of variable-amplitude traveling wave solutions by numerical simulation.See,for example,Figs.2 and 3.

Remark 4 (1) If the real parameter c approaches to zero,then we know that the amplitudes of solutions ui(i=1,2) tend to a constant,therefore all solutions will tend to a special periodic solution.

(2) If we take γ→+∞,then the amplitude of u3will tend to zero,while u4will tend to infinity when λ→+∞.

Figure 1: (a) Traveling wave (2.30) whenTraveling wave (2.30) when

Figure 2: (a) Traveling wave (2.31) when(b) Traveling wave (2.31) when

Figure 3: (a) Traveling wave (2.31) whenTraveling wave (2.31) when

3　Stability Analysis for Individual Traveling Wave Solutions

Setting ρξ=z, (2.3) can be rewritten as the following three-mode dynamical system:

From (2.1) ,a fixed point (ρ0,0,ψ0) of (3.1) corresponds to a plane wave solution of (1.2)

where ξ0is an arbitrary constant.

The individual fixed points of (3.1) may be obtained by substituting z=0 into the right hand sides of (3.1) .This leads to the following equations:

In fact,the constant-amplitude traveling wave solutions ui(i=1,2) obtained in Section 2,are the traveling wave solutions of (1.2) ,that is,they correspond to the following individual fixed points of (3.1) ,

respectively,where

where v and ω satisfies (2.15) .

Remark 5 Actually,using the symmetry, (ρ1,0,ψ1) and (ρ2,0,ψ2) correspond to the same solution u1.Likewise, (ρ3,0,ψ3) and (ρ4,0,ψ4) correspond to the same solution u2.

We now begin to consider the stability of the traveling wave solutions of (1.2) by using the regular phase plane techniques.

By using (3.3) ,it is easy to obtain that the Jacobian matrices of (3.1) at the fixed point Piare of the form

The characteristic equation of the Jacobian matrix at the point of (3.1) may be expressed,after some calculations,as

By using the Routh-Hurwitz conditions,the real parts of all roots of (3.8) are negative if and only if

(3.15) is thus the condition for the stability of the plane wave solution corresponding to the fixed point Pi.

Theorem 3.1 Assume v＞0,ω satisfy (2.15) .Ⅰf Λ＞0 and

then the traveling wave solutions u1,u2are asymptotically stable.

Proof Firstly,if v＞0,then obviously we have δ2＞0.

Secondly,according to (3.6) and (3.15) ,it is easy to obtain (Sω?dQ) (cQ+γS) ＞0.It follows that δ1＞0.

Finally,

Substituting (3.4) and (3.6) , (3.7) and (3.9) into (3.20) ,respectively,we have

It follows that δ3δ2?δ1＞0.The proof is completed.

Theorem 3.2Assume (2.15) or (2.17) is satisfied,also Sv＜0,Q＞0 and cQ+γS＜0,then the traveling wave solutions u1,u2in Section 2 are all unstable.

Proof If Sv＜0,then we have δ3＜0.Meanwhile,if Q＞0 and cQ+γS＜0, it is easy to know δ0＜0.Then there will be at least one eigenvalue with positive real part according to Routh-Hurwitz criterion.

Remark 6 Note that,if a=b=0,then it follows that δ3=0,which means one root of (3.10) will be non-hyperbolic,then the solution will always be unstable, and it will go through a pitch-fork or transcritical bifurcation with subcritical or supercritical.If δ3δ2?δ1=0,then there will be a pair of purely imaginary roots, and a Hopf bifurcation will lead to the onset of periodic solutions of (3.1) .

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(edited by Liangwei Huang)

?This project was supported by the National NSF of China (11571088) ,NSF of Zhejiang Province (LY13A010020) and Program (HNUEYT2013) .

?Manuscript October 31,2015,Revised April 12,2016

?.E-mail:yancongx@163.com