^{1}

^{1}

The (2 + 1)-dimensional Korteweg de Vries (KdV) equation, which was first derived by Boiti et al., has been studied by various distinct methods. It is known that this (2 + 1)-dimensional KdV equation has rich solutions, such as multi-soliton solutions and dromion solutions. In the present article, a unified representation of its N-soliton solution is given by means of pfaffian. We’ll show that this (2 + 1)-dimensional KdV equation is nothing but the Plücker identity when its τ-function is given by pfaffian.

The solitary wave, so-called because it often occurs as a single entity and is localized, was first observed by J. Scott Russell on the Edinburgh-Glasgow Canal in 1834. It is known that many nonlinear evolution equations have soliton solutions, such as the Korteweg de Vries equation, the Sin-Gordon equation, the nonlinear Schrödinger equation, the Kadomtsev-Petviashvili equation, the Davey-Stewartson equation, etc. In order to study the property of nonlinear evolution equations, methods are developed to derive solitary wave solution or soliton solution to nonlinear evolution equations. Some of the most important methods are the inverse scattering transformation (IST) [

In this paper, we are interested in the general expression of N-soliton solution to the (2 + 1)-dimensional KdV equation,

which was first derived by Boiti et al. by using the idea of the weak Lax pair [

In this article, we’ll study the N-soliton solution to the (2 + 1)-dimensional KdV system (1). A compact form of the N-soliton solution to Equation (1) is obtained by means of pfaffian technique, which is given in section 2. Conclusion and further discussions are given in section 3.

Given a nonlinear evolution equation, if it has 3-soliton solution, then this equation is of great possibility of having N-soliton (

Pfaffians are antisymmetric functions with respect to its independent variables

An n-th order pfaffian

where

There are various kinds of pfaffian identities. In this article, we just introduce the so-called Plüker relation for pfaffians [

which we are going to use. Hereafter, we let

The Hirota form of the (2 + 1)-dimensional KdV system (1) is

which is obtained by the dependent variable transformations

Here the Hirota bilinear operator

with n and m are arbitrary nonnegative integers.

In [

where

via the perturbation method. It claims that the N-soliton solutions for

In this article, we’ll study the the multi-soliton solution to Equation (3) using the pfaffian technique [

Proposition 1. If the t-function f of the (2 + 1)-dimensional KdV system (3) is given by the pfaffian function

whose entries, for

then this particular pfaffian function (6) gives an N-soliton solutions to the (2 + 1)-di- mensional KdV system (3).

Proof. In the following, we will prove that the pfaffian function (6) satisfies the (2 + 1)-dimensional KdV Equation (3). By defining “differential operators”

we obtain the following differential formulae

In order to find the pfaffian expression for the differential functions with derivative of variable y, we need to define another letter

Then we have

Substituting formulae (9) and (11) into the right hand side of Equation (3), we obtain nothing but the Plücker relation for pfaffians (2)

where

Note that in order to derive the differential formulae of the pfaffian function (6), we have to define another extra letter

In this article, a compact form of the multi-soliton solution to the (2 + 1)-dimensional KdV system is given via the pfaffian technique. As one can see, the key point of the proof is to derive suitable expressions of the differential formulae of pfaffian t-function f. It is worth pointing out that the method used in this article is different as the one for the proof of the BKP equation, which the differential formulae of the pfaffian t-fun- ction depend only on the “differential operators”

We thank the editor and the referee for their comments. This research work is supported by the National Science Foundation of China (Grant Numbers 11271362, 11271266 and 11501510) and a President’s Grant from the University of Chinese Academy of Sciences. These supports are greatly appreciated.

Zhai, L.X. and Zhao, J.X. (2016) The Pfaffian Technique: A (2 + 1)-Dimensional Korteweg de Vries Equation. Journal of Applied Mathema- tics and Physics, 4, 1930-1935. http://dx.doi.org/10.4236/jamp.2016.410195