Symbolic Computational Study On The Variable-Coefficient Nonliear Models Form Several Fields Such As The Optical Fiber

The nonlinear models play a significant role in many current theoretical studies in several science and engineering fields.They can be used to describe the nonlinear phenomena in the optical fiber,fluid dynamics,solid-state dynamics and plasma physics,etc.By investigating the analytic solutions and integrable properties of the nonlinear models, the essential characteristics of the dynamical mechanism which those nonlinear models describle can be understood much more thoroughly. Recently,considering the inhomogeneity of the media and the uniformity of the boundary,the variable-coefficient nonlinear models are regarded to describe various nonlinear mechanism realistically than the constant-coefficient ones.The computerized symbolic computation possesses the merit that it can operate and realize easily and handle the complex and lengthy expressions together with differential and integral opterations more accurately.The powerful mathematic calculational and graphical functions of symbolic computation software can help us to perform the analytical and observable study on the exact solutions and properties of the variable-coefficient nonlinear models,so that it provides an efficient assistant tool for investigating the variable-coefficient nonlinear models.In this paper,employing symbolic computation,some analytical methods in the soliton theory which can be used to investigate the constant-coefficient nonlinear models are generalized,and then applied to study the variable-coefficient nonlinear models in several fields such as the optial fibers.Using the generalized Painlev(?) analysis,bilinear method, AKNS method,Wronskian technique and Pfaffian method,the exact solutions and integrable properties of the generalized variable-coefficient higher-order nonlinear Schr(o|¨)dinger(HNLS) equation,the variable-coefficient (3+1)-dimensional Kadomstev-Petviashvili(KP)equation,the variable-coeffcient sine-Gordon equation,the inhomogeneous N coupled NLS equation and the variable-coefficient KP equation are analytically investigated.These variable-coefficient nonlinear models can be widely applied to various branches in physics and engineering fields such as the optical fiber,plasma physics,superconduct,fluid dynamics and nonlinear lattice.Specially,they can describe various nonlinear dynamical mechanism in the background with inhomogeneous media and uniform boundary conditions.The main contents of this paper are listed as follows:(Ⅰ) Test the Painlev(?) property for the variable-coefficient nonlinear models based on symbolic computation.The Painlev(?) analysis provides a necessary condition on whether a nonlinear model is completely integrable or not.With the help of symbolic computation,the Painlev(?) analysis is performed on the variable-coefficient(3+1)-dimensional KP equation and the variable-coefficient SG equation.It is found that the variable-coefficient(3+1)-dimensional KP equation does not possess the Painlev(?) property,while the variable-coefficient SG equation is Painlev(?) integrable.(Ⅱ) Generalize the bilinear method and apply the generalization to solve various kinds of exact solutions for several variable-coefficient nonlinear models.In this paper,employing the truncated Painlev(?) expansion or introducing arbitrary parameters into the dependent variable transformation to get the modified one,several variable-coefficient nonlinear models are bilinearized and the solitonic solutions for those models are derived using the formal expansion technique.The main results are as follows:(1) obtain the bright multi-solitonic solutions of the generalized variable-coefficient HNLS equation,and then analyze the influence of the variable coefficients on the soliton properties;(2) transform the generalized variable-coefficient HNLS equation into the constant-coefficient one,obtain the dark solitonic solutions of the generalized variable-coefficient HNLS equation by virtue of the bilinear method and the transformation,and study the stability and propagation proporties of the dark solitons with the help of symbolic and numerical computation;(3) derive the explicit one- and two-solitonic solutions of the inhomogeneous N coupled NLS equation and investigate the properties of soliton propagation;(4) derive the multi-kink solitonic solutions of the variable-coefficient SG equation and analyze the influence of the parameters.(Ⅲ) Investigate the int(?)grable properties such as the B(a|¨)cklund transformation and an infinite number of conservation laws for the variable-coefficient nonlinear models using symbolic computation.In this paper,two different methods are employed to derive the B(a|¨)cklund transformations of the variable-coefficient nonlinear models.Using their bilinearized forms to construct different formal B(a|¨)cklund transformations, the bilinear B(a|¨)cklund transformations of the generalized variable-coefficient HNLS equation,the variable-coefficient SG equation and the inhomogeneous N coupled NLS equation are obtained.Moreover,by the F-Riccati form of the AKNS system for the variable-coefficient nonlinear models,the B(a|¨)cklund transformations inΓfunction form for the variable-coefficient N coupled NLS equation and the variable-coefficient SG equation are derived.Furthermore,an infinite number of conservation laws for the generalized variable-coefficient HNLS equation,the variable -coefficient N coupled NLS equation and the variable-coefficient SG equation are gained using the correspondingΓ-Riccati equations.(Ⅳ) Generalize the Wronskian technique and apply the generalization to obtain the multi-solitonic solutions in Wronski determinant form for variable-coefficient nonlinear models.One of the difficulties to construct the Wronski determinant solution of nonlinear models is finding the properties which the Wronski determinant elements satisfy.Starting from three different viewpoints,the Wronski determinant form multi-solitonic solutions for the variable-coefficient nonlinear models are constructed, and those solutions are analyzed with the help of symbolic computation: (1) construct and verify the Wronski determinant solution for the variable -coefficint(3+1)-dimensional KP equation based on the balancing idea; (2) construct the Wronski determinant solution for the variable-coefficient SG equation using the B(a|¨)cklund transformation for the properties which the determinant elements satisfy;(3) using the AKNS system,construct multi-solitonic solutions in double Wronski determinant form for the generalized variable-coefficient HNLS equation and analyze the properties of solutions.(Ⅴ) Generalize the Pfaffian method and apply to the variable-coefficient nonlinear models.The Pfaffian can be regarded as the generalization of determinant.Using the properties of Pfaffian,the Gramm determinant solution for the variable-coefficient(3+1)-dimensional KP equation are constructed and verified.The Pfaffian procedure which can be used to construct new coupled soliton equations is generalized and applied to the variable-coefficient KP equation.Using the generalization of the Pfaffian procedure,the variable-coefficient coupled KP equations are generated, and then the Wronski- and Gramm-Pfaffian solutions for the coupled variable-coefficient KP equations are derived and verified by the Pfaffian properties.In this paper,the Painlev(?) analysis,bilinear method,AKNS method, Wronskian technique and Pfaffian method are generalized and applied to investigate the solitonic solutions and integrable properties for the variable-coefficient nonlinear models in several fields such as the optical fiber and fluid dynamics using symbolic computation.By virtue of the graphical function of the symbolic computation software,the solitonic solutions of those variable-coefficient nonlinear models are illustrated and their propagation properties are analyzed.It can be expected that these generalized methods in this paper can be used to study a wide class of variable-coefficient nonlinear models in other fields and the analytical research results in this paper for the variable-coefficient nonlinear models in several fields such as the optical fiber can be helpful for further theoretical and experimental researches.