How Many Types of Soliton Solutions do We Know

We discuss several ways of how one could classify the various types of soliton solutions related to NLEE solvable with the generalized n×n Zakharov-Shabat system. In doing that we make use of the fundamental analytic solutions, the dressing procedure and other tools characteristic for the inverse scattering method. We propose to relate to each subalgebra sl(p), 2 ≤ p ≤ n of sl(n), a type of one-soliton solutions which have p−1 internal degrees of freedom.

[1]  V. Zakharov,et al.  Exact Theory of Two-dimensional Self-focusing and One-dimensional Self-modulation of Waves in Nonlinear Media , 1970 .

[2]  V. Gerdjikov.  On the spectral theory of the integro-differential operator a generating nonlinear evolution equations , 1982 .

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

[4]  Mark J. Ablowitz,et al.  Method for Solving the Sine-Gordon Equation , 1973 .

[5]  John Evans,et al.  Nerve Axon Equations: II Stability at Rest , 1972 .

[6]  Allan P. Fordy,et al.  Nonlinear Schrödinger equations and simple Lie algebras , 1983 .

[7]  Leon A. Takhtajan,et al.  Hamiltonian methods in the theory of solitons , 1987 .

[8]  R. Ivanov.  On the dressing method for the generalised Zakharov–Shabat system , 2004, Nuclear Physics B.

[9]  D. Kaup,et al.  Closure of the squared Zakharov--Shabat eigenstates , 1976 .

[10]  Alexey Borisovich Shabat,et al.  Inverse-scattering problem for a system of differential equations , 1975 .

[11]  Vladimir E. Zakharov,et al.  A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I , 1974 .

[12]  S. Novikov,et al.  Theory of Solitons: The Inverse Scattering Method , 1984 .

[13]  V. Gerdjikov,et al.  Completeness of the eigenfunctions for the Caudrey–Beals–Coifman system , 1994 .

[14]  John Evans.  Nerve Axon Equations: III Stability of the Nerve Impulse , 1972 .

[15]  M. Ablowitz,et al.  The Inverse scattering transform fourier analysis for nonlinear problems , 1974 .

[16]  Vladimir E. Zakharov,et al.  Resonant interaction of wave packets in nonlinear media , 1973 .

[17]  D. Kaup.  The solution of the general initial value problem for the full three dimensional three-wave resonant interaction , 1981 .

[18]  V. Zakharov,et al.  Relativistically invariant two-dimensional models of field theory which are integrable by means of the inverse scattering problem method , 1978 .

[19]  S. Helgason.  Differential Geometry, Lie Groups, and Symmetric Spaces , 1978 .

[20]  N. Toyota,et al.  Induced Gauge Structure in Quantum Theory from the Viewpoint of the Confining Potential Approach , 2000 .

[21]  V. Gerdjikov,et al.  N-wave interactions related to simple Lie algebras. ℤ2-reductions and soliton solutions , 2000, nlin/0009034.

[22]  Bäcklund transformations for nonlinear sigma models with values in Riemannian symmetric spaces , 1984 .

[23]  Vladimir S. Gerdjikov,et al.  Generalised Fourier transforms for the soliton equations. Gauge-covariant formulation , 1986 .