Rational Solutions for Lattice Potential KdV Equation and Two Semi-discrete Lattice Potential KdV Equations

Wei Feng; Songlin Zhao; Ying Shi

doi:10.1515/zna-2015-0473

Article

Rational Solutions for Lattice Potential KdV Equation and Two Semi-discrete Lattice Potential KdV Equations

Wei Feng , Songlin Zhao and Ying Shi

Published/Copyright: January 20, 2016

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From the journal Zeitschrift für Naturforschung A Volume 71 Issue 2

Abstract

By imposing reduction conditions on rational solutions for a system involving the Hirota–Miwa equation, rational solutions for lattice potential KdV equation are constructed. Besides, the rational solutions for two semi-discrete lattice potential KdV equations are also considered. All these rational solutions are in the form of Schur function type.

Keywords: Lattice Potential KdV Equation; Rational Solutions; Semi-discrete Lattice Potential KdV Equation.

PACS Number:: 02.30.Ik; 02.30.Ks; 05.45.Yv

Corresponding author: Songlin Zhao, Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310023, P.R. China, E-mail: songlinzhao@zjut.edu.cn

Acknowledgments

This project is supported by the National Natural Science Foundation Grant (Nos 11301483, 11401529, and 11501510).

A Proof of Proposition 1

First, we present the following lemma, which is always needed in Casoratian technique.

Lemma 1 [26] Suppose that B is a N×(N-2) matrix, and a, b, c, and d are Nth-order column vectors, then

(A1)|B, a, b||B, c, d| − |B, a, c||B, b, d| + |B, a, d||B, b, c| = 0. (A1)

Now, we prove Proposition 1. We consider the 123¯ version of equations (2a), i.e.

(A2)(a1 − a2)τ3¯τ1¯2 + (a2 − a3)τ1¯τ2¯3 + (a3 − a1)τ2¯τ1¯3 = 0. (A2)

Multiplying (a₁a₂a₃)^N-2 on the left side of (A2) and noticing the following relations

(A3a)a1N − 2τ1¯ = −|N − 2^, (N − 2)1¯|, (A3a)

(A3b)a2N − 2τ2¯ = −|N − 2^, (N − 2)2¯|, (A3b)

(A3c)a3N − 2τ3¯ = − |N − 2^, (N − 2)3¯|, (A3c)

(A3d)(a1 − a2)(a1a2)N − 2τ1¯2 = |N − 3^, (N − 2)2¯, (N − 2)1¯|, (A3d)

(A3e)(a2 − a3)(a2a3)N − 2τ2¯3 = |N − 3^, (N − 2)3¯, (N − 2)2¯|, (A3e)

(A3f)(a3 − a1)(a1a3)N − 2τ1¯3 = |N − 3^, (N − 2)1¯, (N − 2)3¯|, (A3f)

we get

(A4)(a1a2a3)N − 2[(a1 − a2)τ3¯τ1¯2 + (a2 − a3)τ1¯τ2¯3 + (a3 − a1)τ2¯τ1¯3]= − |N − 3^, (N − 2)2¯, (N − 2)1¯||N − 2^, (N − 2)3¯| − |N − 3^, (N − 2)1¯, (N − 2)3¯||N − 2^, (N − 2)2¯| − |N − 3^, (N − 2)3¯, (N − 2)2¯||N − 2^, (N − 2)1¯|, (A4)

which vanishes in light of Lemma 1, where we employ notations (N-2)_{j ̅}=ϕ_{j ̅}(s+N-2), j=1, 2, 3.

Similarly, we investigate the 123¯ version of (2b) with i=1, j=2:

(A5)(a1 − a2)(ττ1¯2 − τ1¯τ2¯) + τ2¯,xτ1¯ − τ2¯τ1¯,x = 0. (A5)

Noting equalities (A.3a), (A.3b), and (A.3d) together with

a1N − 2τ1¯,x = − |N − 3^, N − 1, (N − 2)1¯| − |N − 2^, (N − 1)1¯|,

a2N − 2τ2¯,x = − |N − 3^, N − 1, (N − 2)2¯| − |N − 2^, (N − 1)2¯|,

then through a direct calculation and noting that Lemma 1 we know that (2b) holds. In terms of the symmetric property, one know that (2b) with i=1, j=3 or i=2, j=3 also holds. Thus, we complete the proof. □

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Received: 2015-11-9

Accepted: 2015-12-15

Published Online: 2016-1-20

Published in Print: 2016-2-1

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Articles in the same Issue

https://doi.org/10.1515/zna-2015-0473

Keywords for this article

Lattice Potential KdV Equation; Rational Solutions; Semi-discrete Lattice Potential KdV Equation.