Tiecheng Xia

A generalized auxiliary equation method and its application to (2+1)-dimensional asymmetric Nizhnik-Novikov-Vesselov equations

A generalized auxiliary equation method is proposed to construct more general exact solutions of nonlinear partial differential equations. With the aid of symbolic computation, we choose the (2+1)-dimensional asymmetric Nizhnik–Novikov–Vesselov equations to illustrate the validity and advantages of the method. As a result, many new and more general exact non-travelling wave and coefficient function solutions are obtained including soliton-like solutions, triangular-like solitions, single and combined non-degenerate Jacobi elliptic doubly-like periodic solutions, and Weierstrass elliptic doubly-like periodic solutions.

The multicomponent generalized Kaup–Newell hierarchy and its multicomponent integrable couplings system with two arbitrary functions

We devise a new simple loop algebra GM and an isospectral problem. By making use of the Tu scheme, the multicomponent generalized Kaup–Newell hierarchy is obtained. Furthermore, an expanding loop algebra FM of the loop algebra GM is presented. Based on FM, the multicomponent integrable couplings system with two arbitrary functions of the multicomponent generalized Kaup–Newell hierarchy are worked out. The method can be applied to other nonlinear evolution equations hierarchy.

Variable-coefficient Jacobi elliptic function expansion method for (2+1)-dimensional Nizhnik–Novikov–Vesselov equations

Abstract In this paper, a variable-coefficient Jacobi elliptic function expansion method is proposed to seek more general exact solutions of nonlinear partial differential equations. Being concise and straightforward, this method is applied to the (2+1)-dimensional Nizhnik–Novikov–Vesselov equations. As a result, many new and more general exact non-travelling wave and coefficient function solutions are obtained including Jacobi elliptic function solutions, soliton-like solutions and trigonometric function solutions. To give more physical insights to the obtained solutions, we present graphically their representative structures by setting the arbitrary functions in the solutions as specific functions.

FROBENIUS INTEGRABLE DECOMPOSITIONS FOR TWO CLASSES OF NONLINEAR EVOLUTION EQUATIONS WITH VARIABLE COEFFICIENTS

Frobenius integrable decompositions are introduced for partial differential equations with variable coefficients. Two classes of partial differential equations with variable coefficients are transformed into Frobenius integrable ordinary differential equations. The resulting solutions are illustrated to describe the solution phenomena shared with the KdV and potential KdV equations, the Boussinesq equation and the Camassa–Holm equation with variable coefficients.

Conservation laws and self-consistent sources for a super KN hierarchy

In the present paper, a super-extension of the Kaup-Newell (KN) hierarchy is proposed by matrix Lie super algebras and the super KN hierarchy with self-consistent sources is established. Furthermore, infinitely many conservation laws of the super integrable hierarchy are presented. The methods derived by us can be generalized to other nonlinear equation hierarchies.

Riemann–Hilbert approach for an initial-boundary value problem of the two-component modified Korteweg-de Vries equation on the half-line

Abstract In this work, we investigate the two-component modified Korteweg-de Vries (mKdV) equation, which is a complete integrable system, and accepts a generalization of 4 × 4 matrix Ablowitz–Kaup–Newell-Segur (AKNS)-type Lax pair. By using of the unified transform approach, the initial-boundary value (IBV) problem of the two-component mKdV equation associated with a 4 × 4 matrix Lax pair on the half-line will be analyzed. Supposing that the solution {u1(x, t), u2(x, t)} of the two-component mKdV equation exists, we will show that it can be expressed in terms of the unique solution of a 4 × 4 matrix Riemann–Hilbert problem formulated in the complex λ-plane. Moreover, we will prove that some spectral functions s(λ) and S(λ) are not independent of each other but meet the global relationship.

The multi-component dispersive long wave equation hierarchy, its integrable couplings and their Hamiltonian structures

Abstract An isospectral problem is established based on loop algebra X ∼ , by making use of the generalized Tu scheme the multi-component Liouville integrable dispersive long wave (DLW) equation hierarchy is obtained. Then, two expanding loop algebra F ∼ M and Y ∼ are presented, which devoted to working out two integrable couplings of the multi-component DLW equation hierarchy. Finally, the Hamiltonian structures of the multi-component DLW equation hierarchy and the second integrable couplings are obtained by employing the trace variational identity.

Nonlinear integrable couplings of a generalized super Ablowitz-Kaup-Newell-Segur hierarchy and its super bi-Hamiltonian structures

In this paper, a new generalized

A generalized new auxiliary equation method and its applications to nonlinear partial differential equations

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A generalized F-expansion method and new exact solutions of Konopelchenko–Dubrovsky equations

matrix spectral problem of Ablowitz-Kaup-Newell-Segur(AKNS) type associated with the enlarged matrix Lie super algebra is proposed and its corresponding super soliton hierarchy is established. The super variational identities is used to furnish super-Hamiltonian structures for the resulting super soliton hierarchy.