Shoufeng Shen

Lie symmetry analysis of the time fractional KdV-type equation

Abstract The Lie symmetry analysis method is extended to deal with the time fractional KdV-type equation. It is shown that this equation can be reduced to an equation with the Erdelyi–Kober fractional derivative. This method can also be applied to symmetry classification of fractional equations with some arbitrary functions.

An integrable generalization of the Kaup-Newell soliton hierarchy

A generalization of the Kaup–Newell spectral problem associated with sl is introduced and the corresponding generalized Kaup–Newell hierarchy of soliton equations is generated. Bi-Hamiltonian structures of the resulting soliton hierarchy, leading to a common recursion operator, are furnished by using the trace identity, and thus, the Liouville integrability is shown for all systems in the new generalized soliton hierarchy. The involved bi-Hamiltonian property is explored by using the computer algebra system Maple.

A note on a class of Hardy-Rellich type inequalities

In this note we provide simple and short proofs for a class of Hardy-Rellich type inequalities with the best constant, which extends some recent results.MSC:26D15, 35A23.

A generalized Dirac soliton hierarchy and its bi-Hamiltonian structure

Abstract By using symbolic computation software(Maple), a generalized Dirac soliton hierarchy is derived from a new matrix spectral problem associated with the Lie algebra sl ( 2 , R ) . A bi-Hamiltonian structure yielding Liouville integrability is furnished by the trace identity.

Bäcklund Transformations and Solutions of Some Generalized Nonlinear Evolution Equations

Using the method of separation of variables based on Backlund transformation, new classification results with exact solutions for various generalized nonlinear evolution equations such as the KdV-type equations, the NNV-type equations and the Schrodinger-type equations are obtained. We also present a unified construction method for nonlinear evolution equations which admit separation of variable and the main tool is a class of generalized multidimensional binary Bell polynomials.

A Soliton Hierarchy Associated with a Spectral Problem of 2nd Degree in a Spectral Parameter and Its Bi-Hamiltonian Structure

Associated with , a new matrix spectral problem of 2nd degree in a spectral parameter is proposed and its corresponding soliton hierarchy is generated within the zero curvature formulation. Bi-Hamiltonian structures of the presented soliton hierarchy are furnished by using the trace identity, and thus, all presented equations possess infinitely commuting many symmetries and conservation laws, which implies their Liouville integrability.

A class of third-order nonlinear evolution equations admitting invariant subspaces and associated reductions

With the aid of symbolic computation by Maple, a class of third-order nonlinear evolution equations admitting invariant subspaces generated by solutions of linear ordinary differential equations of order less than seven is analyzed. The presented equations are either solved exactly or reduced to finite-dimensional dynamical systems. A number of concrete examples admitting invariant subspaces generated by power, trigonometric and exponential functions are computed to illustrate the resulting theory.

A modified complex short pulse equation of defocusing type

In this paper, we are concerned with a modified complex short pulse (mCSP) equation of defocusing type. Firstly, we show that the mCSP equation is linked to a complex coupled dispersionless equation of defocusing type via a hodograph transformation, thus, its Lax pair can be deduced. Then the bilinearization of the defocusing mCSP equation is formulated via dependent variable and hodograph transformations. One- and two-dark soliton solutions are found by Hirota’s bilinear method and their properties are analyzed. It is shown that, depending on the parameters, the dark soliton solution can be either smoothed, cusponed or looped one. More specifically, the dark soliton tends to be evolved into a singular (cusponed or looped) one due to the increase of the spatial wave number in background plane waves and the increase of the depth of the trough. In the last part of the paper, we derive the defocusing mCSP equation from the single-component extended KP hierarchy by the reduction method. As a by-product, the N-dark soliton solution in the form of determinants for the defocusing mCSP is provided.

A symmetry classification algorithm of the generalized differential–difference equations

Abstract In this letter, a symmetry classification algorithm of the generalized differential–differenceequations u n = F n t , u n − 1 , u n , u n + 1 , u n − 1 , u n + 1 is proposed according to the intrinsic Lie point symmetries, allowed transformations and Lie algebraic structures. The validity of this approach is demonstrated by a Toda-type lattice.