Wronskian $\pmb {N}$-soliton solutions to a generalized KdV equation in ($\pmb {2+1}$)-dimensions

Abstract

The aim of this paper is to construct Wronskian solutions to a generalized KdV equation in ($2+1$)-dimensions, which possesses a trilinear form. On the basis of two useful properties associated with Hirota differential operators, a general Wronskian formulation is established and the involved functions for Wronskian entries satisfy a system of combined linear partial differential equations. The key technique is to apply the Wronskian identity of the bilinear KP equation while presenting those sufficient conditions. Other illustrative examples of sufficient conditions are also given for the cKP3-4 equation, the ($2+1$)-dimensional DJKM equation, and the dissipative ($2+1$)-dimensional AKNS equation. Finally, N-soliton solutions and soliton molecules are worked out through the presented Wronskian formation.

Appendix

The proof of Lemma 2.1 is as follows:

Proof

By applying the conditions (2.5) and differential rules for determinants as defined in [18], we have the following derivatives of the Wronskian determinant $f=f_{N}=|\widehat{N-1}|$:

$$\begin{aligned} \frac{\partial f}{\partial y}=(a_1\partial _{x}+a_2\partial _x^2+\dots +a_{m}\partial _x^{m}) {f} \end{aligned}$$

(A.1a)

$$\begin{aligned} \equiv (a_{1}\partial _{x_1}+a_{2}\partial _{x_2}+\dots +a_{m}\partial _{x_m})f , \end{aligned}$$

(A.1b)

$$\begin{aligned} \frac{\partial f}{\partial t}=(b_1\partial _{x}+b_2\partial _x^2+\dots +b_{n}\partial _x^{n}) {f}\end{aligned}$$

(A.1c)

$$\begin{aligned} \equiv (b_{1}\partial _{x_1}+b_{2}\partial _{x_2}+\dots +b_{n}\partial _{x_n})f . \end{aligned}$$

(A.1d)

Let us next suppose that

(A.2)

where $\delta _{nr}, r=1,2,\ldots ,\mathrm {C_{m+n-1}^{n}},$ are expansion coefficients. Using the definition of D-operators and the above conditions (A.1), we further get

(A.3)

It means that Lemma 2.1 holds.