N -soliton solutions and the Hirota conditions in (2+1)-dimensions
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N‑soliton solutions and the Hirota conditions in (2+1)‑dimensions Wen‑Xiu Ma1,2,3,4 Received: 13 September 2020 / Accepted: 21 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We compute N-soliton solutions and analyze the Hirota N-soliton conditions, in (2+1)-dimensions, based on the Hirota bilinear formulation. An algorithm to check the Hirota conditions is proposed by comparing degrees of the polynomials generated from the Hirota function in N wave vectors. A weight number is introduced while transforming the Hirota function to achieve homogeneity of the resulting polynomial. Applications to three integrable equations: the (2+1)-dimensional KdV equation, the Kadomtsev–Petviashvili equation, the (2+1)-dimensional Hirota–Satsuma–Ito equation, are made, thereby providing proofs of the existence of N-soliton solutions in the three model equations. Keywords N-Soliton solution · Hirota N-soliton condition · (2+1)-Dimensional integrable equations Mathematics Subject Classification 35Q51 · 35Q53 · Secondary · 37K40
1 Introduction It is known that N-soliton solutions are a kind of exact solutions to weakly nonlinear dispersive wave equations (Ablowitz and Segur 1981; Novikov et al. 1984). Particularly, solitons superimposed in fibers can be applied to optical communications (Hasegawa 1989 and 1990). Many interesting solutions in mathematical physics, including breather, peakon, complexiton, lump and rogue wave solutions, are special reductions of N-soliton solutions in different situations. The Hirota bilinear method is a standard and powerful technique to present N-soliton solutions (Hirota 2004). The innovative concept of bilinear derivatives is a generalized idea to deal with nonlinear differential equations, and through bilinear forms, it becomes much more natural to generate N-soliton solutions. * Wen‑Xiu Ma [email protected] 1
Department of Mathematics, Zhejiang Normal University, Jinhua 321004, Zhejiang, China
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Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia
3
Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620, USA
4
School of Mathematics, South China University of Technology, Guangzhou 510640, China
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Hirota bilinear derivatives are defined by Hirota (2004): ( ) m ∑ m−i m Dm f ⋅ g = (−1) (𝜕xi f )(𝜕xm−i g), x i i=1
(1.1)
m ≥ 1,
and more generally, bilinear partial derivatives with multiple variables are similarly defined: n m n � � (Dm x Dt f ⋅ g)(x, t) = (𝜕x − 𝜕x� ) (𝜕t − 𝜕t� ) f (x, t)g(x , t )|x� =x,t� =t ,
m, n ≥ 1.
(1.2)
m ≥ 1,
(1.3)
m, n ≥ 1.
(1.4)
When f = g , we obtain Hirota bilinear expressions:
D2m−1 f ⋅ f = 0, D2m x x f ⋅f =
( ) 2m ∑ 2m (−1)2m−i (𝜕xi f )(𝜕x2m−i f ), i i=1
and similarly, bilinear partial derivative expressions: ( )( ) m n ∑ ∑ n j n−j m n m+n−i−j m Dx Dt f ⋅ f = (−1) (𝜕xi 𝜕t f )(𝜕xm−i 𝜕t f ), i j i=1 j=1
By means of Hirota bilinear expressions, we can define Hirota bilinear equations. Take an even
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