Abstract
In this paper, we seek connections between the Sylvester equation and Kadomtsev–Petviashvili system. By introducing Sylvester equation LM are bold, please chekc if bold neceaasry, if not, please remove all bold of equation −MK = rsT together with an evolution equation set of r and s, master function S(i,j)=sTKjC(I + MC)−1Lir is used to construct the Kadomtsev–Petviashvili system, including the Kadomtsev–Petviashvili equation, modified Kadomtsev–Petviashvili equation and Schwarzian Kadomtsev–Petviashvili equation. The matrix M provides τ-function by τ = |I + MC|. With the help of some recurrence relations, the reductions to the Korteweg–de Vries and Boussinesq systems are discussed.
Highlights
The Sylvester equation [1–3]Citation: Feng, W.; Zhao, S
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As a by-product of the direct linearization (DL) method, the Cauchy matrix (CM) approach was first proposed by Nijhoff and his collaborators to investigate soliton solutions of the Adler–Bobenko–Suris lattice list except for the elliptic case of Q4
Summary
As a by-product of the DL method, the CM approach was first proposed by Nijhoff and his collaborators to investigate soliton solutions of the Adler–Bobenko–Suris lattice list except for the elliptic case of Q4 This scheme proves to be a powerful tool with the ability to provide integrable equations together with their soliton solutions, and amongst others, Lax representation, Bäcklund and Miura transformations. With K, M ∈ C N × N , r, s ∈ C N ×1 , and some continuous integrable equations, including the KdV, modified KdV, Schwarzian KdV and sine-Gordon equations They showed that all these equations arose from the Sylvester Equation (2) and could be expressed by some discrete equations of master function S(i,j) = sT K j ( I + M )−1 K i r,. We have an Appendix A as a compensation of the paper
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