An algebraic method to construct the Darboux matrix

We present an effective procedure to construct the 1‐soliton Darboux matrix. Our approach, based on the Zakharov–Shabat–Mikhailov’s dressing method, is especially useful in the case of non‐canonical normalization and for non‐isospectral linear problems. The construction is divided into two steps. First, we represent a given linear problem as a system of some algebraic constraints on two matrices. In this context we introduce and discuss invariants of the Darboux matrix. Second, we derive the Darboux matrix demanding that it preserves the algebraic constraints. In particular, we consider in details the restrictions imposed by various reduction groups on the form of the Darboux matrix.

[1]  D. Levi,et al.  N-solitons on a vortex filament☆ , 1983 .

[2]  P. Gragert,et al.  Exact solution to localized-induction-approximation equation modeling smoke ring motion. , 1986, Physical review letters.

[3]  K. Konno,et al.  Some Remarkable Properties of Two Loop Soliton Solutions , 1983 .

[4]  J. Cieśliński.  Non-isospectral deformations of the Heisenberg ferromagnet equation , 1990 .

[5]  A. Sym.  Soliton surfaces , 1983 .

[6]  Vladimir E. Zakharov,et al.  Inverse scattering method with variable spectral parameter , 1987 .

[7]  M. Lakshmanan,et al.  Geometrical and gauge equivalence of the generalized Hirota, Heisenberg and Wkis equations with linear inhomogeneities , 1985 .

[8]  On the geometry of the inhomogeneous Heisenberg ferromagnet: nonintegrable case , 1993 .

[9]  A. Degasperis,et al.  Exact solution via the spectral transform of a generalization with linearlyx-dependent coefficients of the nonlinear Schrödinger equation , 1978 .

[10]  J. Cieśliński.  An effective method to compute N‐fold Darboux matrix and N‐soliton surfaces , 1991 .

[11]  A. Sym.  Soliton surfaces , 1984 .

[12]  Orlando Ragnisco,et al.  Extension of the Zakharov-Shabat generalized inverse method to solve differential-difference and difference-difference equations , 1980 .

[13]  F. R. Hama,et al.  Localized‐Induction Concept on a Curved Vortex and Motion of an Elliptic Vortex Ring , 1965 .

[14]  Decio Levi,et al.  A hierarchy of coupled Korteweg-de Vries equations and the normalisation conditions of the Hilbert-Riemann problem , 1983 .

[15]  Gernot Neugebauer,et al.  Einstein-Maxwell solitons , 1983 .

[16]  V. Matveev,et al.  Darboux transformation and explicit solutions of the Kadomtcev-Petviaschvily equation, depending on functional parameters , 1979 .

[17]  R. Balakrishnan.  On the inhomogeneous Heisenberg chain , 1982 .

[18]  V. Zakharov,et al.  On the integrability of classical spinor models in two-dimensional space-time , 1980 .

[19]  D. Levi,et al.  Soliton surfaces , 1985 .

[20]  D. Levi,et al.  Dressing methodvs. classical Darboux transformation , 1984 .

[21]  A. Mikhailov,et al.  The reduction problem and the inverse scattering method , 1981 .

[22]  M. Lakshmanan,et al.  Geometry of generalised nonlinear Schrödinger and Heisenberg ferromagnetic spin equations with linearly x-dependent coefficients , 1980 .

[23]  A. Sym,et al.  On integrability of the inhomogeneous Heisenberg ferromagnet model: Examination of a new test , 1994 .

[24]  R. Meinel,et al.  General N-soliton solution of the AKNS class on arbitrary background , 1984 .

[25]  D. Levi,et al.  Integrable systems describing surfaces of non-constant curvature , 1990 .