Exact Solutions of the Nonlocal Nonlinear Schrödinger Equation with a Perturbation Term

1 Introduction

The nonlinear Schrödinger equation [1], [2], [3], [4], [5], [6], [7], [8], [9],

has been used to characterise rogue waves generated by nonlinear energy transfer in an open ocean [2] and broadband optical pulse propagation in nonlinear fibres [3], where i²=−1, ∗ is the complex conjugate, while μ is the envelope of the wave field and depends on the variables ς and ζ [3]. Rogue waves and solitons are both the analysis solutions of (1) with different seed solutions. Physically speaking, a rogue wave is thought of as an isolated huge wave with an amplitude claimed to be much larger than the average wave crests around it in the ocean [4], [5], [6] and also seen in other fields [2], [7], [6]. A soliton is a solitary wave that preserves its velocity and shape after an interaction [4]; i.e. the soliton can be considered as a quasi-particle [8], [9].

Besides, nonlinear Schrödinger equation with a perturbation term has also been studied [10]:

where B is a real constant and P is a complex function of the real variables ι and κ. The trivial condensate solution of (2) is P=B. While this solution is unstable with respect to small perturbations, the growth rate of instability is pB2−p2/4, where p is the wave number of perturbation. Relations of rogue waves and the nonlinear stage of modulation instability for (2) have been attained via the dressing method [10].

On the other hand, nonlocal nonlinear Schrödinger equation have been discussed [11]:

where Q(X, T) is the complex-valued function of the real variables X and T. Solutions of (3) have been attained by virtue of the inverse scattering transform [11], [12], [13], [14], [15]. It is remarkable that (3) turns out to be Lax integrable [12], [13], [14], [15]. Some properties of (3) has also been proved as follows [11]: time-reversal symmetry: if Q(X, T) is a solution, so is Q^*(X, −T); invariance under the transformation X→−X: If Q(X, T) is a solution, so is Q(−X, T); PT symmetry: if Q(x, t) is a solution, so is Q^*(−X, −T); gauge invariance: if Q(X, T) is a solution, so is exp(iθ₀)Q(X, T) with real and constant θ₀; complex translation invariance: If Q(X, T) is a solution, so is Q(X+iX₀, T) for any constant and real X₀.

In this paper, we will work on the the nonlocal nonlinear Schrödinger equation with a perturbation,

where q(x, t) is a complex function of the real variables x and t and A is a real constant. We note that (4) also has the same properties as (3) as mentioned above; q(x, t)=A is a condensate solution of (4).

The paper is organised as follows. By virtue of the symbolic computation [16], [17], [18], [19], [20], in Section 2, Lax pairs, Darboux transforms, and N-order solutions of (4) will be obtained. In Section 3, trilinear forms of (4) will be attained. In Section 3 are the conclusions.

2 Lax Pairs and Darboux Transformation of (4)

Lax pairs could transform a nonlinear evolution equation into a linear system. Thus, via the lax pairs, N-order solutions of (4) can be obtained. By virtue of the Ablowitz-Kaup-Newell-Segur [21] procedure, Lax pairs of (4) are

where Φ(x, t)=[ϕ₁₁(x, t), ϕ₁₂(x, t)]^T is a vector function of x and t, superscript T denotes the transposition for the vector/matrix, U and V are both the matrix functions of x and t as follows:

U=[−λq(x, t)−q∗(−x, t)λ],V=[i[−q∗(−x, t)q(x, t)−A2]−2iλ22iλq(x, t)−iq(x, t)x2iλ−q∗(−x, t)−iq∗(−x, t)x2iλ2−i[−q∗(−x, t)q(x, t)−A2]],

where λ is the complex spectrum parameter. Via the compatibility condition, U_t−V_x+UV−VU=0, (4) can be attained.

3 Trilinear Forms of (4)

Multi-linear forms of the nonlinear evolution equations can help people to attain their solutions, especially to the bilinear forms of those [22]. In the following, we will obtain the trilinear forms of (4) by virtue of the independent variable substitution. With the aid of the transformation,

where f(x, t) is a real function of the variables x and t, while g(x, t) is a complex function of the variables x and t, (4) have the following trilinear forms:

where ?is a complex constant and the operator DxmDtn is defined as [22]

with m and n both as the integers, a(x, t) as the function of x and t, and b(x′, t′) as the function of formal variables x′ and t′.

4 Conclusions

In this paper, we have studied the nonlocal nonlinear Schrödinger equation (4) with a perturbation term. Our main results are displayed as follows: (a) Trilinear forms (3) of (4) have been attained, by virtue of the independent variable transformation. (b) Lax pairs (5) and Darboux transformation (Section 2.1) have been both obtained. Via the Darboux transformation, two kinds of the N-order solutions of (4) with different seed solutions have been attained. From the above discussion, we have found that some properties of (4) are different from both (3) and (2).