Discrete and Semidiscrete Models for AKNS Equation

Song-lin Zhao; Ying Shi

doi:10.1515/zna-2016-0443

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Discrete and Semidiscrete Models for AKNS Equation

Song-lin Zhao und Ying Shi

Veröffentlicht/Copyright: 31. Januar 2017

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Abstract

We establish a discrete model for the Ablowitz–Kaup–Newell–Segur (AKNS) equation via generalised Cauchy matrix approach. Two semidiscrete AKNS equations are obtained by, respectively, introducing straight continuum limit and skew continuum limit. Some reductions are also discussed.

Keywords: Continuum Limit; Discrete AKNS Equation; Reduction; Semidiscrete AKNS Equation; Solutions

PACS: 39A14; 35Q51; 37K40

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11301483

Award Identifier / Grant number: 11401529

Award Identifier / Grant number: 11371241

Award Identifier / Grant number: 11501510

Funding source: Natural Science Foundation of Zhejiang Province

Award Identifier / Grant number: LY17A010024

Funding statement: This project is supported by the National Natural Science Foundation of China (Grant Nos. 11301483, 11401529, 11371241, and 11501510) and the Natural Science Foundation of Zhejiang Province (Grant No. LY17A010024).

A List of notations

In the following, we first list some notation, where the subscripts _D and _J correspond to the cases of matrices being diagonal and being Jordan block, respectively.

Diagonal matrix:
(A.1)ΓD[N]({ki}1N)=Diag(k1, k2, …, kN),
Jordan block matrix:
(A.2)ΓJ[N](a)=(a00⋯001a0⋯0001a⋯00⋮⋮⋮⋮⋮⋮000⋯1a)N × N,
Lower triangular Toeplitz matrix: ^[1]
(A.3)T[N]({aj}1N)=(a100⋯00a2a10⋯00a3a2a1⋯00⋮⋮⋯⋮⋮⋮aNaN − 1aN − 2⋯a2a1)N × N,
Skew triangular Hankel matrix:
(A.4)H[N]({bj}1N)=(b1⋯bN − 2bN − 1bNb2⋯bN − 1bN0b3⋯bN00⋮⋮⋮⋮⋮bN⋯000)N × N.

Besides the above mentioned matrix notations we used, let us list other needed ones below.

(A.5a)discrete plane wave factor: ρi=(p+kip−ki)n(q+kiq−ki)mρi0, with constants ρi0,

(A.5b)discrete plane wave factor: σj=(p−ljp+lj)n(q−ljq+lj)mσj0, with constants σj0,

(A.5c)Nth order vector: r1,D[N]({kj}1N)=(ρ1, ρ2, ⋯, ρN)T,

(A.5d)Nth order vector: r1,J[N](k1)= (ρ1, ∂k1ρ11!, ⋯, ∂k1N − 1ρ1(N−1)!)T,

(A.5e)Nth order vector: r2,D[N]({lj}1N)=(σ1, σ2, ⋯, σN)T,

(A.5f)Nth order vector: r2,J[N](l1)=(σ1, ∂l1σ11!, ⋯, ∂l1N − 1σ1(N−1)!)T,

(A.5g)2N×2N matrix: GDD[N1;N2]({ki}1N1; {lj}1N2) =(0GDD;12[N1;N2]({ki}1N1; {lj}1N2)−GDD;12[N1;N2]T({ki}1N1; {lj}1N2)0), GDD;12[N1;N2]({ki}1N1; {lj}1N2)=(1ki−lj)N1 × N2,

(A.5h)2N×2N matrix: GDJ[N1;N2]({ki}1N1; b) =(0GDJ;12[N1;N2]({ki}1N1; b)−GDJ;12[N1;N2]T({ki}1N1; b)0), GDJ;12[N1;N2]({kj}1N1; b)=(gi,j)N1 × N2, gi,j=(1ki−b)j,

(A.5i)2N×2Nmatrix: GJD[N1;N2](a; {lj}1N2) =(0GJD;12[N1;N2](a; {lj}1N2)−GJD;12[N1;N2]T(a; {lj}1N2)0), GJD;12[N1;N2](a; {lj}1N2)=(gi,j)N1 × N2, gi,j=−(−1a−lj)i,

(A.5j)2N×2Nmatrix: GJJ[N1;N2](a; b) =(0GJJ;12 [N1;N2](a; b)−GJJ;12[N1;N2]T(a; b)0), GJJ;12[N1;N2](a; b) =(gi,j)N1 × N2, gi,j=Ci + j − 2i − 1(−1)i + 1(a−b)i + j − 1,

where

Cji=j!i!(j−i)!, (j≥i).

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Received: 2016-11-16

Accepted: 2016-12-27

Published Online: 2017-1-31

Published in Print: 2017-3-1

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Artikel in diesem Heft

https://doi.org/10.1515/zna-2016-0443

Schlagwörter für diesen Artikel

Continuum Limit; Discrete AKNS Equation; Reduction; Semidiscrete AKNS Equation; Solutions