quad a linear system. In order to better understand

quad Most real world complex natural phenomenons are described and abstracted by a nonlinear model, it is difficult to truly reflect the real world by a linear system. In order to better understand various behaviour of the real world problems, to seek solutions of nonlinear partial differential equation and to study the characteristics of its solution (e.g. soliton properties) researchers looking for soliton solution of nonlinear differential equations by using different methods, such as variables separation method, inverse scattering method, B”{a}cklund and Darboux transformation method, (G/G$^{prime}$) expansion method, bifurcation method, Hirota bilinear method and Lie group method and other methods.

The following sections give an introduction of background, present situation and progress for some of the aforementioned methods cite{ZB08}.subsection{Variables separation method}label{chapt:2.1.

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1}Variables separation and Fourier transform method are the two widely used methods of linear wave theory in solution determination, but these two methods cannot be directly used in nonlinear system. Around 1960s, both methods has been very successfully extended to integrable nonlinear model, known as the inverse scattering method. Until recent developments researchers do a lot of very outstanding work on the separation variable method and the extended separation variable method with geometric method and Ansatz method cite{Doly96, ZRF93}.From the relation of reflectionless potential and eigenfunction of the soliton equation, in 1989, Cao Ze-we formulated an efficient method for constructing finite dimensional integrable Hamiltonian systems from infinite dimensional integrable systems cite{Cao90}. This is put forward by researchers for the first time in the form of separation of variables method cite{CGW99, CL91}, then called nonlinearization method or Lax-pair nonlinearization method.

The non-linearization approach decomposes soliton equations into temporal and spatial finite-dimensional Hamiltonian systems, which also makes it very natural to compute solutions of soliton equations numerically, for instance, by the symplectic method. Recently, researchers study and apply this method in finding a soliton solution for different models to list a few cite{ZM99a, ZM99b, MG00, MC07, QZ97}.In order to achieve the true sense of separation of variables, in 1996, Lou and Lu  cite{LL96}, studied the approach on dynamical system and put forward a method of separation of variables, this is the prototype of the linear separation of variables. In 1999, Lou and Chen also used, this method to confirm that the separation of variables method only applicable to Lax integrable system cite{LC99}. Five years latter, researchers gradually established a perfect linear separation of variable method and make multiple linear separation variable method to develop so that can promote real nonlinear model for a large number of equations .So far, more than linear separation variable method has successfully solved the categories of (1+1)–dimensional nonlinear systems and some (2+1)–dimensional, (3+1)–dimensional and nonlinear systems and have been successfully applied to the differential difference system cite{Lou02, TLZ02, TL03}.

There are two very important nonlinear variable separation method that are functional separation variable method and derivative related functional separation variable method. Zhdanov (1995) put forward a new method of separation of variables called functional separation variable method cite{Zhda95} and Qu Changzheng and et al studied further the functional separation solution of nonlinear evolution equations by using classified general conditions and solve  the steps and realize the method to obtain functional solution of separation of variables of (1+1)– dimensional, (2+1)–dimensional and general nonlinear diffusion equation cite{QZL00,QE04}.Recently, Zhang Shunli and et al, has been applied a functional separation variable method to general nonlinear diffusion equation, general KdV equation and general nonlinear wave equation which are made a complete separation of variables cite{ZL04, ZLQ03}. Separation of variables is the most common approach to solve linear equations of mathematical physics. Despite progress in nonlinear method of separation of variables, in many ways, but still not perfect, there are many important areas to be studied, particularly in derivative-related functional method of separation of variables cite{Zhar09}.subsection{Inverse — Scattering method}label{chapt:2.

1.2}quad Most of the hardest problems and interesting phenomena being studied by mathematicians, engineers and physicists are nonlinear in nature. The classical inverse scattering method was invented during investigation of the KdV equation by use of quantum mechanics and Schr”{o}dinger operator of eigenvalue problem  cite{KdV95}. Thus, successful analysis has usually depended on an ability to decompose the PDEs into a set of ODEs.

Most integrable PDEs uses as a cornerstone the backscatter method (or inverse scattering method) refinements cite{ZS72, AKNS74} through the systematic use of a Lax pair cite{Lax68}.quad Lax induces the backscatter method into a more refined mathematical form and make it become a kind of soliton equation (or system) and points out the premise of the equation by inverse scattering method to find the equation of Lax. In 1973, Zakharov and Shabat cite{ZS73} has successfully applied the ideas of Lax, that is a backscatter method, to the nonlinear Schr”{o}dinger  equation of initial value problems and the Lax method is extended to the matrix form which can be viewed in retrospect as the beginning of the study of integrable PDEs on metric graphs cite{Gard67}. Case and Kac cite{Case73} extended the inverse scattering method to discrete version and used to solve discrete Schr”{o}dinger equation.quad In 1974 Satsuma and Yajima cite{SY74} generalize the inverse scattering method and used the method to other NLEEs for finding a soliton solution. In 1975 and 1976 Wahlquist and Estabrook put forward a method for systematically deriving a set of interrelated potentials and pseudo–potentials for nonlinear PDEs in two independent variables of a prolongation cite{WE75, EW76}. It provides a necessary conditions for solving the equation by inverse scattering method with the aid of Lie algebra.quad Flaschka cite{Flas74} and Ablowitz and et al cite{AML75} respectively, studied the discrete backscatter method to exponential Toda Lattice of initial value problem and the nonlinear dual mesh equation.

As application of this method, the general N–soliton formula is derived and constants of the motion are expressed interms of the scattering data. In 1983, Ke Wu, Hanying Guo and Shikun Wang studied the inverse scattering method for solving the Lax pair of given NLEEs to reduced a kind of Riemann–Hilbert (RH) problem  of meromorphic function with respect to the complex spectral parameter cite{KHS83, SHK83}. They established a special relationship between the balcklund transformation and the inverse scattering method.quad Aktosum cite{Akot91} studied the method of inverse scattering method to solve the limited area on the nonlinear Schr”{o}dinger equation. Fokas and Its studied this method which is generalized to half infinite region cite{Foka04}.

The rapid development of science and technology today inverse scattering method is still coruscate gives unlimited vitality, has been widely applied to geophysical prospective, super–symmetry in the fields of quantum mechanics.quad In 2004, Ablowitz, Prinair and Trubatch studied the inverse scattering method that allows one to linearize a class of NLEEs. In doing so, they obtain global information about the structure of the solution cite{Ablo04}.

In 2015, Caudrelier reflect in his article a framework to solve open problem by formulating inverse scattering method for an integrable PDE on a star–graph  cite{CaV15}. He used the nonlinear Schr”{o}dinger equation to extend the unique method of Fokas to initial value problem of a matrix.quad Liu and et al, in 2016, use inverse scattering method for studying derivative nonlinear Schr”{o}dinger equation on the line using its gauge equivalence with a related nonlinear dispersive equation cite{LPS16}. They prove Lipschitz continuity of the direct and inverse scattering maps from the weighted Sobolev spaces $H_{(2,2)}(mathbb{R})$ to itself. And they found the global existence of solutions to the derivative nonlinear Schr”{o}dinger equation for initial data.subsection{B”{a}cklund and Darboux transform method}label{chapt:2.

1.3}quad B”{a}cklund transformations have emerged over the past decade as an important tool in the study of a wide range of NPDEs in mathematical physics. It have been applied in several branches of physics and mathematics, some of them are for the treatment of finite amplitude dispersive waves in ultrashort optical pulse propagation, the nonlinear heat conduction, in the nonlinear dielectric medium and in general relativity cite{ADD02}. These transformations allowed us to construct nonlinear superposition principles and use this technique to obtain Multisoliton solutions for many important NLEEs cite{Draz89}.In mathematics, B”{a}cklund transformation is typically a system of first order PDEs relating two functions and often depending on an additional parameter. A B”{a}cklund transform which relates solutions of the same equation is called an invariant B”{a}cklund transform or auto–B”{a}cklund transform.

If such a transform can be found, much can be deduced about the solutions of the equation especially if the B”{a}cklund transform contains a parameter cite{DB76}. In 1973 Wahlquist and Estabrook cite{WE73, WE75}, studied the KdV–equations which have a B”{a}cklund transformation and its modified form to obtain a Pfaffian system which will generate a new solution of the KdV equation from any given solution.In 1986 Chern and K. Tenenblat cite{CT86}, investigated properties of evolution equations in terms of pseudospherical surfaces. Moreover, they discuss a geometrical result which, under certain conditions, provides B”{a}cklund transformations for differential equations which describe pseudospherical surfaces. This idea was extended to the sine-Gordon equation by Sasaki cite{Sasa79}.

A real valued function of a real variable is said to be a Darboux transformation, if it maps every connected set in its domain onto a connected set. Bruckner A–M, Bruckner J–B studied Darboux transformations to generalize and summarize the application in different models (see cite{BB67}, the reference there in).In 1975 Wadati, Sanuki and Konno cite{WSK74}, studied B”{a}cklund and Darboux transformation together for exact solutions of nonlinear wave equation that has made great development and they studied the relationships among the three methods, that is, inverse scattering method, B”{a}cklund transformation and an infinite number of conservation laws on KdV equation, mKdV equation and sine–Gordon equation. Ablowitz, Kaup, Newell and Segur (AKNS) studied the Lax mKdV equations to extend to more general situations of AKNS system cite{AKN73}.

Gu Chaohao, Hu cite{Gu92,GHZ04} apply the Darboux transformation to a high–dimensional AKNS system, echelon KdV cite{GH86}, echelon mKdV–sine–Gordon cite{Gu86}.B”{a}cklund and Darboux transformation were used to mean curvature surface and projective space to construct the harmonic mapping on differential geometry cite{Hu84, Hu99}. In 1995, Darboux studied Schr”{o}dingers equation to find a general solution using a Darboux transformation and he used a seed solution of nonlinear equations and its Lax–Pair solution to obtain nonlinear equations cite{Boli95}. In addition, many researchers did a lot of good work in the field of B”{a}cklund transformation and the development of Darboux transformation to list a few cite{HW87, Zull13, AM13, Raha15}.