An Integrable Nonlinear Wave Model: Darboux Transform and Exact Solutions

: Soliton equations are inﬁnite-dimensional integrable systems described by nonlinear partial differential equations. In the mathematical theory of soliton equations, the discovery of integrability of these equations has greatly promoted the understanding of their generality


Introduction
The discovery of many solitons and integrable systems and the in-depth study of their mathematical and physical properties is one of the great advances in nonlinear science, and it has been applied in a series of scientific and technological fields. A lot of mathematical theories, such as symplectic manifold, spectral theory of differential operators, partial differential equations, Lie algebras and Lie groups and their representation theory, algebraic curves and so on, have become important tools in the study of soliton theory. In turn, the research progress of soliton theory promotes the development of these subdisciplines. Several systematic approaches have been proposed to solve these integrable nonlinear equations, for example, the inverse scattering transform [1], algebraic curve method [2,3,4,12], Darboux transform [5,6,7,8], and other methods [9,10,11]. By using these methods, explicit exact solutions to many integrable nonlinear equations are constructed, including soliton solutions, quasiperiodic solutions, breather solutions, rogue-wave solutions, peakon solutions, etc [13,14,15,16,17,18,19,20].
Darboux transform (DT) is a very useful tool for solving integrable nonlinear equations. It can be used to generate new solutions from various known solutions. Furthermore, this process may proceed continuously, usually resulting in a series of exact solutions. In this article, one proposes a new integrable nonlinear wave model related to a 2 × 2 matrix spectral problem and an integrable reduction (w = w * , v = u * ) Then their DTs are constructed by resorting to the introduced gauge transforms between the Lax pairs and the reduction technique. As an illustrative example of applying DT, some explicit exact solutions of the integrable nonlinear reduction equation (2) are obtained, such as solitons, breathers and rogue waves.
The present paper is organized as follow. In Section 2, one first derives a Lax pair of the nonlinear wave equation (1) and then deduces a gauge transform between the Lax pairs, from which the DT of the nonlinear wave model (1) is constructed. Using the reduction technique, the DT of the integrable nonlinear reduction (2) is gained. In Section 3, appropriate parameters are chosen to get the corresponding "seed solutions". Various explicit exact solutions of the integrable nonlinear reduction (2), such as solitons, breathers and rogue waves, are given by using the DT.

Darboux Transforms
In the present section, one will find a Lax pair of the nonlinear wave equation (1) and construct its DT. Then, a DT of the integrable nonlinear reduction (2) is obtained through the reduction technique. For this purpose, one introduces a Lax pair, a 2 × 2 matrix spectral problem and an auxiliary problem, where u, v, w are three potentials, and λ ∈ C is a spectral parameter independent with x and t. By direct calculation, the following conclusion holds. Theorem 2.1. Assume that φ satisfies (3) and (4). Then the compatibility condition φ xt = φ tx generates the zero-curvature equation, U t − V x + [U, V ] = 0, which is exactly the nonlinear wave equation (1).
Assuming α 1 , α 2 , . . . , α N ∈ C are some arbitrary constants, a general solution of the spectral problems (3) and (4) at λ j is given by ϕ(λ j ) + α j ψ(λ j ). Therefore, one considers the following linear algebraic system: with The constants λ j and α j are chosen such that the coefficient matrix of (9) is nondegenerate.
, are uniquely given by (9). It's easy to verify from (9) that which means that λ j (1 ≤ j ≤ 2N ) are the roots of det T . Because det T = 1 when λ = 0, det T can be written as The above results show that λ = λ j (1 ≤ j ≤ 2N ) are removable isolated singularities ofÛ andV . Therefore,Û andV for all λ ∈ C can be defined by analytic continuations. If a gauge transform changes a Lax pair into another Lax pair of the same type, one calls it a DT of the integrable nonlinear equation related to the Lax pair.
Theorem 2.2. The Lax matrixÛ given by (7)-(10) possesses the same form as the Lax matrix Û where new potentialsû,v,ŵ are determined by the DT (8).
Theorem 2.4. Let (u, v, w) be a solution of the nonlinear wave model (1). Then, (û,v,ŵ) determined by the DT (8) is a new solution of the nonlinear wave model (1), where A k , B k , C k , D k of (8) are uniquely given by the linear algebraic system (9).
Assume that A straightforward calculation shows that where A k and B k are determined by For example, when N = 1 and when N = 2, one obtains from (32), respectively, that and Then, one arrives at the following result. Theorem 2.5. Let (u, v) be a solution of the integrable reduction model (2). Assume the DT readŝ where B 1 and A k , (1 ≤ k ≤ N ), are determined by the system of linear algebraic equations (32). Then (û,ŵ) given by the DT (35) is a new solution of the integrable reduction model (2).

Conclusions
In the present paper, one introduces a 2 × 2 matrix spectral problem and derives a new integrable nonlinear wave system. The system is in itself interesting and is simplified to a novel integrable complex nonlinear wave equation. One finds the gauge transform between the Lax pairs of the integrable nonlinear wave system and constructs its Darboux transforms. Through the reduction technique, Darboux transforms of the integrable nonlinear reduction equation are obtained by analysing the symmetries of the Lax pair. On this basis, an algebraic algorithm is given to solve the integrable nonlinear wave system and its integrable nonlinear reduction. As an illustrative example of our method, some explicit exact solutions of the integrable nonlinear reduction equation are constructed by resorting to the resulting Darboux transform and Mathematica software, including soliton solutions, breather solutions, rogue-wave solutions. These results are very convenient for application and analysis. In addition to this, one knows even less about the integrable nonlinear wave system and its integrable nonlinear reduction. Whether the integrable nonlinear wave system and its integrable nonlinear reduction have Bäcklund transform, conserved quantity, Hamiltonian structure and other properties is still a problem to be solved and will be discussed later.