Study Of Multi-Linear Variable Separation And Localized Excitation In Soliton Systems

Finding explicit and exact solutions, in particular, solitary wave solutions, of soliton systems in mathematical physics play an important role in soliton theory. It is well known that the multi-linear variable separation approach(MLVSA) has been one of powerful methods to construct solutions for a broads class of considerable physically interesting and important nonlinear evolution equations. The independent variables of a reduced field in the multi-linear variable separation approach have totally been separated. In this paper, we study the multi-linear variable separation approach and give its key steps to solve several physically significant soliton systems. The paper is organized as follows:Part I is devoted to introducing some techniques to realize the the multi-linear variable separation approach in soliton systems, such as the homogenous balance approach, the Painleve truncated expansion approach, the general mapping deformation method and the direct Backlund transformation method. These techniques are employed respectively to explore the variables separation solutions of (2+l)-dimensional soliton systems, such as Burgers equation, BKK equation, GKdV equation, HBK equation, DLW equation, BLP equation, MNNV equation, sG equation. It is worth to mention that these techniques are valid for some other physically significant (2+l)-dimensional models apart from ones mentioned in this paper. Selecting some types of seed solutions for the sake of including as many arbitrary functions as possible is a important procedure. Different forms of solution can be obtained from different separation forms of variable. For instance, we can obtain three different types of variable separated solutions for the (2+1)-dimensional sG(MNNV) equation, which owe something to the seed solutions. In fact, we can obtain the same variable separation solutions via the homogenous balance approach and the Painleve truncated expansion approach, though they have different particular steps. The variable separated solutions via the general map-ping deformation method is the special case of the solutions via the above-mentioned two kinds of techniques. And, the direct Backlund transformation method require the models to have very good symmetry property. Thanks to the complicated structure of the (2+l)-dimensional soliton systems, there is quite abundant solutions remaining to be announced. One can choose suitable techniques above to solve the concrete nonlinear evolution models effectively. In addition, the more arbitrary functions included in the variable separated form, the more the form is close to the generally solution. Therefore, it is the goal in our study to add the arbitrary functions as much as possible, for instance, there are four arbitrary functions in the variable separated form of (2+l)-dimensional sG(MNNV) equation.It is also quite important to investigate the obtained variable separation solutions, which have several arbitrary functions(p and q). Part II deals with some special types of excitations of (2+l)-dimensional sG(MNNV) equation via some suitable selections of p and q. In addition to the many types of stable localized excitations such as dromions, lumps, breathers, instantons, ring solitons, horseshoe like solitons and foldons, there may be many types of chaotic and fractal patterns by selecting the arbitrary functions as the chaotic and fractal solutions of some lower dimensional non-integrable models. Certainly, if we choose p and q as periodic functions, we will get the nonlocal periodic wave patterns. In addition, in some models, p is an arbitrary function while q should be satisfied a Riccati equation (qt = a(i) + b(t)q + c(t)q2) or heat conductive equation^ = (qaqy)y), such as (2+l)-dimensional Burgers equation. These constraints make the Burgers equation have no saddle types of ring solitons and analytical lump solutions while these types of excitations do exist for the other equations.The interaction between solitons of the (2+l)-dimensional nonlinear models may have quite rich phenomena. On the other hand, the interaction in higher spatial dimensions continues to be much more intricate because it is very difficulty to solve higher dimensions nonlinear models. Part III brings up that the appearance of the arbitrary functions is closely related to the arbitrary boundary conditions of some types of the quantities(or potentials) for the related models. One can investigate the stability properties of the solutions presented in this paper and their relevance as asymptotic states for suitable initial boundary value problems. Therefore, one could select the necessary and sufficient conditions of the arbitrary functions for the completely elastic interaction.A nearly systematic process of the multilinear variable separation approach to solve (2+l)-dimensional soliton systems has been established, however, it is difficult to apply multi-linear variable separation approach to the (l+l)-dimensional nonlinear models. In part IV, we further investigate the extension of the multi-linear variable separation...