Mei Jian-Qin

Lie Algebras and Integrable Systems

A 3 × 3 matrix Lie algebra is first introduced, its subalgebras and the generated Lie algebras are obtained, respectively. Applications of a few Lie subalgebras give rise to two integrable nonlinear hierarchies of evolution equations from their reductions we obtain the nonlinear Schrodinger equations, the mKdV equations, the Broer-Kaup (BK) equation and its generalized equation, etc. The linear and nonlinear integrable couplings of one integrable hierarchy presented in the paper are worked out by casting a 3 × 3 Lie subalgebra into a 2 × 2 matrix Lie algebra. Finally, we discuss the elliptic variable solutions of a generalized BK equation.

New Families of Soliton and Periodic Solutions of Bose?Einstein Condensation in Linear Magnetic Field and Time-Dependent Laser Field

By using a new generally projective Riccati equation method and with the help of symbolic computation, we consider a nonlinear Gross–Pitaevskii equation with weak bias magnetic and time-dependent laser fields. As a result, some new soliton solutions, rational function solution, and periodic solutions are obtained.

Symmetry Reductions and Group-Invariant Solutions of (2+1)-Dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada Equation

With the aid of symbolic computation, we present the symmetry transformations of the (2 + 1)-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada equation with Lous direct method that is based on Lax pairs. Moreover, with the symmetry transformations we obtain the Lie point symmetries of the CDGKS equation, and reduce the equation with the obtained symmetries. As a result, three independent reductions are presented and some group-invariant solutions of the equation are given.

Group Classification and Exact Solutions of a Class of Variable Coefficient Nonlinear Wave Equations

Complete group classification of a class of variable coefficient (1 + 1)-dimensional wave equations is performed. The possible additional equivalence transformations between equations from the class under consideration and the conditional equivalence groups are also investigated. These allow simplification of the results of the classification and further applications of them. The derived Lie symmetries are used to construct exact solutions of special forms of these equations via the classical Lie method. Nonclassical symmetries of the wave equations are discussed.

New Solitary Solutions of (2+1)-Dimensional Variable Coefficient Nonlinear Schrödinger Equation with an External Potential

By a series of transformations, the (2+1)-dimensional variable coefficient nonlinear Schrodinger equation can turn to the Klein–Gordon equation. Many new double travelling wave solutions of the Klein–Gordon equation are obtained. Thus, the new solitary solutions of the variable coefficient nonlinear Schrodinger equation with an external potential can be found.

Noether-Type Symmetries and Associated Conservation Laws of Some Systems of Nonlinear PDEs

The algorithm for constructing conservation laws of Euler–Lagrange–type equations via Noether-type symmetry operators associated with partial Lagrangian has been presented. As applications, many new conservation laws of some important systems of nonlinear partial differential equations have been obtained.