Levi, D.; Sym, A.; Wojciechowski, S. A hierarchy of coupled Korteweg-de Vries equations and the normalization conditions of the Hilbert-Riemann problem. (English) Zbl 0548.35098 J. Phys. A 16, 2423-2432 (1983). The paper studies the Zakharov-Shabat reduction [V. E. Zakharov and A. B. Shabat, Funkts. Anal. Prilozh. 13, No.3, 13-22 (1979; Zbl 0448.35090)] of the integration of solvable nonlinear equation to the solution of a matrix Hilbert-Riemann problem: for a closed contour \(\Gamma\) on the compact complex \(\lambda\) -plane and \(N\times N\) matrix function G defined on \(\Gamma\) one has to determine the factorization \(G(\lambda)=\psi_ 1(\lambda)\psi_ 2(\lambda)\), where \(\psi_ i\), \(N\times N\) matrix functions of \(\lambda\) defined on \(\Gamma\), are to be extended analytically respectively inside and outside \(\Gamma\). The authors discuss the normalization conditions \(\psi_ 2(\lambda_ 0)=\chi\), where \(\lambda_ 0\in {\mathbb{C}}\) and \(\chi\) is some fixed \(N\times N\) matrix, which are of importance for constructing of soliton solutions. The normalization matrix \(\chi\) which depends, in principle, through the soliton fields and on the independent variables x and t can be (under some reasonable conditions) gauge reduced to the constant matrix. The coupled Korteweg-de Vries equations are considered as an example of the developed technique and the soliton solution is calculated. Reviewer: V.Z.Enol’skij Cited in 1 ReviewCited in 20 Documents MSC: 35Q99 Partial differential equations of mathematical physics and other areas of application 35Q15 Riemann-Hilbert problems in context of PDEs Keywords:Zakharov-Shabat reduction; solvable nonlinear equation; matrix Hilbert- Riemann problem; normalization conditions; soliton; coupled Korteweg-de Vries equations Citations:Zbl 0448.35090 PDFBibTeX XMLCite \textit{D. Levi} et al., J. Phys. A, Math. Gen. 16, 2423--2432 (1983; Zbl 0548.35098) Full Text: DOI