Commutativity of Pfaffianization and Bäcklund transformations: the KP equation

Abstract

The Pfaffianization procedure is applied to the modified KP equation. As a result, we derive a coupled modified KP system, which is shown to constitute a bilinear Bäcklund transformation for the coupled KP (cKP) equation introduced by Hirota and Ohta. This means that commutativity of Pfaffianization and Bäcklund transformation is valid for the KP equation. As a by-product, we explicitly derive a coupled system from Bäcklund transformation formulae of the cKP equation, which solves an open problem proposed by Hirota. In addition to starting from the Bäcklund transformation, a Lax pair for the coupled KP equation is worked out. Finally, two kinds of Pfaffian solutions to the coupled modified KP equation are derived.

Multiple skew orthogonal polynomials and 2-component Pfaff lattice hierarchy

“…This is the bilinear form of the so-called modified coupled KP equation, which plays an important role in the study of commutativity of Pfaffianization and Bäcklund transformation [33].…”

On the integrability of the (1+1)‐dimensional and (2+1)‐dimensional Ito equations

“…As is well known, bilinear representation, bilinear Bäcklund transformation (BT), Lax pair, and multi‐soliton solutions are very important properties of many integrable systems. Among the methods, which are used to research on this topic, the Hirota bilinear method is applied most widely till now . The key of Hirota bilinear method is through a dependent variable transformation to put the original nonlinear evolution equation (NLEE) into a form where the new dependent variable(s) appears bilinear.…”

Darboux covariant Lax pairs and infinite conservation laws of the (2+1)-dimensional breaking soliton equation

“…It relies on a particular skill in using appropriate exchange formulas which are connected with the linear representation of the system. [1][2][3][4] However, in recent years Lambert and co-workers have proposed a procedure to obtain parameter families of bilinear Bäcklund transformation for soliton equations in a lucid and systematic way based on the use of Bell polynomials. [5][6][7] The Bell polynomials are found to play an important role in the characterization of bilinearizable equations.…”