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In this work we establish the traveling wave solution with parameters of the KdV-Sawada-Kotera equation by using the G^'G-expansion method. This method is effectively employed. When the parameters are taken as special values, solitary, periodic and rational solutions are obtained.

The simple direct method is adopted to find Non-Auto-Backlund transformation for variable coefficient non-linear systems. The (2+1)-dimensional generalized Kadomtsev-Petviashvili equation with variable coefficients is used as an example to elucidate the solution procedure, and its symmetry transformation and exact solutions are obtained.

By using some exact solutions of the auxiliary ordinary differential equation, a direct algebraic method is obtained to construct the exact complex solutions for non-linear partial differential equations. The method is implemented for the (2+1)-dimensional BKP equation. New exact complex solutions are obtained.

Starting with the solution classification for a linear differential equations, the complexiton solutions to nonisospectral Korteweg–de Vries equation are presented. The basic technique adopted is the Wronskian technique for Hirota's bilinear equation.

Darboux transformation and Backlund transformation in bilinear form for the nonisospectral KP equation are first investigated. Corresponding solutions are derived by using the Backlund transformation in bilinear form. It has been shown that these transformations are auto-Backlund transformations for isospectral problems while not for nonisospectral ones.

A new 4 × 4 isospectral problem with three potentials and the corresponding hierarchy of nonlinear evolution equations are presented. Especially, a new coupled KdV equation is produced. Their generalized bi-Hamiltonian structures are also investigated by using the trace identity. Moreover, a new finite-dimensional Hamiltonian system is given through the nonlinearization of the corresponding Lax pair. Enough conserved integrals, which are in involution and functionally independent, are created by the Lax operator to guarantee Liouville integrability of the Hamiltonian system.

Multi-soliton solutions to the modified nonlinear Schr?dinger equation (MNLSE) with variable coefficients (VCs) in inhomogeneous fibers are obtained with the help of mapping transformation, which reduces the VC MNLSE into a constant-coefficient MNLSE. Based on the analytical solutions, one- and two-soliton transmissions in the proper dispersion management systems are discussed. The sustainment of solitons and the disappearance of breathers for the VC MNLSE are first reported here.