Analytical study on a two-dimensional Korteweg–de Vries model with bilinear representation, Bäcklund transformation and soliton solutions

Abstract

With symbolic computation, Bell-polynomial scheme and bilinear method are applied to a two-dimensional Korteweg–de Vries (KdV) model, which is firstly proposed with Lax pair generating technique. Bell-polynomial expression with one auxiliary independent variable is derived and transformed into bilinear form. According to the coupled two-field conditions between the primary and replica fields, Bell-polynomial-typed Bäcklund transformations (BTs) are constructed and converted into the bilinear ones. Finally, soliton solutions of the two-dimensional KdV model are obtained (via solving the bilinear representation and BT, respectively) and compared. Such associated integrable properties as bilinear representation, BT (especially auxiliary-independent-variable-involved Bell-polynomial-typed ones constructed in this paper) and soliton solutions (especially the multi-soliton ones) may be useful for further study on other two-dimensional KdV and KdV-typed models.