Generalised Fourier transform and perturbations to soliton equations (original) (raw)
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Basic aspects of soliton theory
2006
This is a review of the main ideas of the inverse scattering method (ISM) for solving nonlinear evolution equations (NLEE), known as soliton equations. As a basic tool we use the fundamental analytic solutions (FAS) of the Lax operator L. Then the inverse scattering problem for L reduces to a Riemann-Hilbert problem. Such construction cab be applied to wide class
Quantum perturbation theory of solitons
Nuclear Physics B, 1975
We develop a systematic perturbation theory for amplitudes involving a single soliton (extended finite energy solution of a classical field theory). These methods allow one in principle to compute to any desired order quantum corrections to soliton masses, form factors and Green functions. The major technical problem encountered is an infrared divergence of the naive perturbation theory rules. This divergence is dominated by a modification of the usual functional methods so as to properly define the soliton as a momentum eigenstate. The extension of these methods to soliton-soliton scattering is discussed.
Inverse Scattering Transform and the Theory of Solitons
Encyclopedia of Complexity and Systems Science, 2009
Time evolution of the scattering data The evolvement of the scattering data from its initial value S(λ, 0) at t = 0 to its value S(λ, t) at a later time t. I. Definition of the Subject and Its Importance A general theory to solve NPDEs does not seem to exist. However, there are certain NPDEs, usually first order in time, for which the corresponding IVPs can be solved by the IST method. Such NPDEs are sometimes referred to as integrable evolution equations. Some exact solutions to such equations may be available in terms of elementary functions, and such solutions are important to understand nonlinearity better and they may also be useful in testing accuracy of numerical methods to solve such NPDEs. Certain special solutions to some of such NPDEs exhibit particle-like behaviors. A single-soliton solution is usually a localized disturbance that retains its shape but only changes its location in time. A multi-soliton solution consists of several solitons that interact nonlinearly when they are close to each other but come out of such interactions unchanged in shape except for a phase shift. Integrable NPDEs have important physical applications. For example, the KdV equation is used to describe [14,23] surface water waves in long, narrow, shallow canals; it also arises [23] in the description of hydromagnetic waves in a cold plasma, and ion-acoustic waves in anharmonic crystals. The nonlinear Schrödinger (NLS) equation arises in modeling [24] electromagnetic waves in optical fibers as well as surface waves in deep waters. The sine-Gordon equation is helpful [1] in analyzing the magnetic field in a Josephson junction (gap between two superconductors).
From single- to multiple-soliton solutions of the perturbed KdV equation
The solution of the perturbed KdV equation (PKDVE), when the zero-order approximation is a multiplesoliton wave, is constructed as a sum of two components: elastic and inelastic. The elastic component preserves the elastic nature of soliton collisions. Its perturbation series is identical in structure to the series-solution of the PKDVE when the zero-order approximation is a single soliton. The inelastic component exists only in the multiple-soliton case, and emerges from the first order and onwards. Depending on initial data or boundary conditions, it may contain, in every order, a plethora of inelastic processes. Examples are given of sign-exchange soliton-anti-soliton scattering, soliton-anti-soliton creation or annihilation, soliton decay or merging, and inelastic soliton deflection. The analysis has been carried out through third order in the expansion parameter, exploiting the freedom in the expansion to its fullest extent. Both elastic and inelastic components do not modify soliton parameters beyond their values in the zero-order approximation. When the PKDVE is not asymptotically integrable, the new expansion scheme transforms it into a system of two equations: The Normal Form for ordinary KdV solitons, and an auxiliary equation describing the contribution of obstacles to asymptotic integrability to the inelastic component. Through the orders studied, the solution of the latter is a conserved quantity, which contains the dispersive wave that has been observed in previous works.
Spectral Theory of Solitons on a Generic Background for the KPI Equation
The article reports on two lectures given at the Workshop. In the case of solutions of the KPI equation, non vanishing along a nite number of directions at large distances, a reformulation of the inverse scattering method in terms of a sort of complex extension of the resolvent of the related Lax operator is presented. Jost solutions for the continuum and discrete spectrum are constructed. The analytic properties and a bilinear representation of the resolvent are given. The study of scattering data and of their characterization properties is postponed to a following paper. The reader interested in more details on the resolvent approach proposed in this paper can consult ref. 1 { 7 and for previous works on the scattering transform for the KPI equation ref. 8 { 12 .
Methods and Applications of Analysis, 2000
A theory for constructing integrable couplings of soliton equations is developed by using various perturbations around solutions of perturbed soliton equations being analytic with respect to a small perturbation parameter. Multi-scale perturbations can be taken and thus higher dimensional integrable couplings can be presented. The theory is applied to the KdV soliton hierarchy. Infinitely many integrable couplings are constructed for each soliton equation in the KdV hierarchy, which contain integrable couplings possessing quadruple Hamiltonian formulations and two classes of hereditary recursion operators, and integrable couplings possessing local 2 + 1 dimensional bi-Hamiltonian formulations and consequent 2 + 1 dimensional hereditary recursion operators.
Inverse Scattering Transform, KdV, and Solitons
Current Trends in Operator Theory and its Applications, 2004
In this review paper, the Korteweg-de Vries equation (KdV) is considered, and it is derived by using the Lax method and the AKNS method. An outline of the inverse scattering problem and of its solution is presented for the associated Schrödinger equation on the line. The inverse scattering transform is described to solve the initial-value problem for the KdV, and the time evolution of the corresponding scattering data is obtained. Soliton solutions to the KdV are derived in several ways.