Zhengde Dai

Applications of HTA and EHTA to YTSF Equation

New approaches, homoclinic test approach (HTA) and extended homoclinic test approach (EHTA), are proposed to seek solitary-wave solution of high dimensional nonlinear wave system. Exact periodic solitary-wave, periodic soliton, cross solitary-wave and doubly periodic wave solutions for YTSF equation are obtained using HTA and EHTA, respectively.

Exact three-wave solutions for the KP equation

A new type of three-wave solution, periodic two-solitary-wave solutions, for (1+2)D Kadomtsev-Petviashvili (KP) equation is obtained using the extended three-soliton method and with the help of Maple.

Exact three-wave solution for higher dimensional KdV-type equation

A new periodic type of three-wave solutions including periodic two-solitary solution, doubly periodic solitary solution and breather type of two-solitary solution for the (1+2)-dimensional and (1+3)-dimensional KdV-type equations are obtained using Hirotas bilinear form and generalized three-wave type of ansatz approach.

Exact soliton solutions for the fifth-order Sawada–Kotera equation

Exact soliton solutions for the fifth-order Sawada–Kotera equation are obtained by using the Hirota bilinear method. These solutions include one-soliton solutions, periodic two-soliton solutions and singular periodic soliton solutions. The results show that there exist periodic two-soliton solutions and singular periodic soliton solutions for the fifth-order Sawada–Kotera equation.

Exact three-wave solutions for the (3+1)-dimensional Jimbo-Miwa equation

Exact three-wave solutions including periodic cross-kink wave solutions, doubly periodic solitary wave solutions and breather type of two-solitary wave solutions for the (3+1)-dimensional Jimbo-Miwa equation are obtained using the generalized three-wave method. It is shown that the generalized three-wave method, with the help of symbolic computation, provides an effective and powerful mathematical tool for solving high-dimensional nonlinear evolution equations in mathematical physics.

Exact periodic cross-kink wave solutions and breather type of two-solitary wave solutions for the (3 + 1)-dimensional potential-YTSF equation ☆

Abstract In this paper, the (3+1)-dimensional potential-YTSF equation is investigated. Exact solutions with three-wave form including periodic cross-kink wave, periodic two-solitary wave and breather type of two-solitary wave solutions are obtained using Hirota’s bilinear form and generalized three-wave approach with the aid of symbolic computation. Moreover, the properties for some new solutions are shown with some figures.

The EHTA for nonlinear evolution equations

Abstract Two types of important nonlinear evolution equations are investigated by using the extended homoclinic test approach (EHTA). Some exact soliton solutions including breather type of soliton, periodic type of soliton and two soliton solutions are obtained. These results show that the extended homoclinic test technique together with the bilinear method is a simple and effective method to seek exact solutions for nonlinear evolution equations.

New Two-soliton and Periodic Solutions to KdV Equation

There has been an active and exciting search for explicit solutions of nonlinear evolution equations(NLEES) ever since the discovery of the soliton in 1834.Many method have been proposed over these years for finding these solutions [1-10] including a number of asymptotic methods that have been proposed recently by J.He [11,12] and the double-exp function method [13] for finding the exact multi-wave solutions and double-wave solutions of NLPDEs. In this work, we show that the most elementary ansatz of double Exp-function method can be produced by an extension of two-soliton ansatz in a fractional form. Applying this ansatz to KdV equation, new abundant two-solitary solutions and periodic solution are obtained. We consider KdV equation [1]:

Extend three-wave method for the (1+2)-dimensional Ito equation

Abstract In this work, Extend three-wave method (ETM) is used to construct the novel multi-wave solutions of the (1+2)-dimensional Ito equation. As a result, three-soliton solution, doubly periodic solitary wave solutions, periodic two solitary wave solutions are obtained. It is shown that the Extend three-wave method may provide us with a straightforward and effective mathematical tool for seeking multi-wave solutions of higher dimensional nonlinear evolution equations.

The breather-type and periodic-type soliton solutions for the (2+1 )-dimensional breaking soliton equation

Exact breather-type and periodic-type soliton solutions including the double-breather-type soliton solutions, the breather-type periodic soliton solutions and breather-type two-soliton solutions, and the periodic-type two-soliton and three-soliton solutions for the (2+1)-dimensional breaking soliton equation are obtained using the extended three-wave method (ETM). The results show that the ETM may provide us with a straightforward and effective mathematical tool for seeking multi-wave solutions of higher dimensional nonlinear evolution equations.