Vandermonde-like determinants’ representations of Darboux transformations and explicit solutions for the modified Kadomtsev–Petviashvili equation

✍ Scribed by Ding-jiang Huang; Hong-qing Zhang

• Publisher: Elsevier Science
• Year: 2008
• Tongue: English
• Weight: 645 KB
• Volume: 387
• Category: Article
• ISSN: 0378-4371
• DOI: 10.1016/j.physa.2008.03.020

✦ Synopsis

Recently, the (2 + 1)-dimensional modified Kadomtsev-Petviashvili (mKP) equation was decomposed into two known (1 + 1)-dimensional soliton equations by Dai and Geng [H.H. Dai, X.G. Geng, J. Math. Phys. 41 (2000) 7501]. In the present paper, a systematic and simple method is proposed for constructing three kinds of explicit N-fold Darboux transformations and their Vandermonde-like determinants' representations of the two known (1 + 1)-dimensional soliton equations based on their Lax pairs. As an application of the Darboux transformations, three explicit multi-soliton solutions of the two (1 + 1)dimensional soliton equations are obtained; in particular six new explicit soliton solutions of the (2 + 1)-dimensional mKP equation are presented by using the decomposition.

The explicit formulas of all the soliton solutions are also expressed by Vandermonde-like determinants which are remarkably compact and transparent.