A systematic method is developed which allows one to identify certain important classes of evolution equations can be solved by the inverse scattering. The form each equation characterized dispersion relation its associated linearized version and an integro‐differential operator. comprehensive presentation scattering given general features solution are discussed. relationship theory Backlund transformations brought out. In view role relation, comparatively simple asymptotic states,...

We present the inverse scattering method which provides a means of solution initial-value problem for broad class nonlinear evolution equations. Special cases include sine-Gordon equation, sinh-Gordon Benney-Newell Korteweg-de Vries modified and generalizations.

We develop here two aspects of the connection between nonlinear partial differential equations solvable by inverse scattering transforms and ordinary (ODE) P-type (i.e., no movable critical points). The first is a proof that solution an ODE, obtained solving linear integral equation certain kind, can have any points. second algorithm to test whether given ODE satisfies necessary conditions be P-type. Often, used or not evolution may completely integrable.

The conceptual analogy between Fourier analysis and the exact solution to a class of nonlinear differential–difference equations is discussed in detail. We find that dispersion relation associated linearized equation prominent developing systematic procedure for isolating solving equation. As examples, number new are presented. method makes use techniques inverse scattering. Soliton solutions conserved quantities worked out.

The initial value problem for the sine-Gordon equation is solved by inverse-scattering method.Received 7 March 1973DOI:https://doi.org/10.1103/PhysRevLett.30.1262©1973 American Physical Society

A method is presented which enables one to obtain and solve certain classes of nonlinear differential−difference equations. The introduction a new discrete eigenvalue problem allows the exact solution self−dual network equations be found by inverse scattering. has as its singular limit continuous Zakharov Shabat. Some interesting differences arise both in scattering analysis time dependence from previous work.

A new integrable nonlocal nonlinear Schrödinger equation is introduced. It possesses a Lax pair and an infinite number of conservation laws PT symmetric. The inverse scattering transform data with suitable symmetries are discussed. method to find pure soliton solutions given. An explicit breathing one solution found. Key properties discussed contrasted the classical equation.

Two-dimensional lump solutions which decay to a uniform state in all directions are obtained for the Kadomtsev–Petviashvili and two-dimensional nonlinear Schrödinger type equation. The amplitude of these is rational its independent variables. These constructed by taking ’’long wave’’ limit corresponding N-soliton direct methods. describing multiple collisions lumps also presented.

Complex variables provide powerful methods for attacking problems that can be very difficult to solve in any other way, and it is the aim of this book a thorough grounding these their application. Part I text provides an introduction subject, including analytic functions, integration, series, residue calculus also includes transform methods, ODEs complex plane, numerical methods. II contains conformal mappings, asymptotic expansions, study Riemann–Hilbert problems. The authors extensive...

A nonlocal nonlinear Schrödinger (NLS) equation was recently introduced and shown to be an integrable infinite dimensional Hamiltonian evolution equation. In this paper a detailed study of the inverse scattering transform NLS is carried out. The direct problems are analyzed. Key symmetries eigenfunctions data conserved quantities obtained. theory developed by using novel left–right Riemann–Hilbert problem. Cauchy problem for formulated methods find pure soliton solutions presented; leads...

A nonlinear partial difference equation is obtained and solved by the method of inverse scattering. In a certain continuum limit it shown how this approximates Schrodinger related differential‐difference equation. At all times solutions can be compared, scheme to convergent. These ideas apply other evolution equations as well.

We consider the evolution of packets water waves that travel predominantly in one direction, but which wave amplitudes are modulated slowly both horizontal directions. Two separate models discussed, depending on whether or not long comparison with fluid depth. These two-dimensional generalizations Korteweg-de Vries equation (for waves) and cubic nonlinear Schrödinger short waves). In either case, we find depends fundamentally dimensionless surface tension particular, for waves,...

A nonlocal nonlinear Schrödinger (NLS) equation was recently found by the authors and shown to be an integrable infinite dimensional Hamiltonian equation. Unlike classical (local) case, here nonlinearly induced “potential” is symmetric thus NLS also symmetric. In this paper, new reverse space‐time time equations are introduced. They arise from remarkably simple symmetry reductions of general AKNS scattering problems where nonlocality appears in both space or alone. dynamical systems. These...

Rational solutions of certain nonlinear evolution equations are obtained by performing an appropriate limiting procedure on the soliton direct methods. In this note specific attention is directed at Korteweg–de Vries equation. However, methods used quite general and apply to most with isospectral property, including multidimensional equations. latter case, nonsingular, algebraically decaying, can be constructed.

There is a connection between nonlinear partial differential equations that can be solved by the inverse scattering transform and ordinary without movable critical points (e.g., Painlev\'e transcendents). We exploit this to reduce second equation of linear integral equation. also describe class exactly linearized method.

The initial value problem of the Kadomtsev‐Petviashvili equation for one choice sign in has been recently investigated literature. Here we consider other sign. We introduce suitable eigenfunctions which though bounded are not analytic spectral parameter. This, contrast to known case, prevents us from formulating inverse as a nonlocal Riemann‐Hilbert boundary problem. Nevertheless formulation is given and formal solution constructed via linear integral equation.

It is known through the inverse scattering transform that certain nonlinear differential equations can be solved via linear integral equations. Here it demonstrated ’’directly,’’ i.e., without Jost-function formalism solution of equation actually solves equation. In particular, this extends scope methods to ordinary which are found Painlevé type. Some global properties these ODE’s obtained rather easily by approach.

An exactly solvable discrete $PT$ invariant nonlinear Schr\"odinger-like model is introduced. It an integrable Hamiltonian system that exhibits a nontrivial symmetry. A one-soliton solution constructed using left-right Riemann-Hilbert formulation. shown this pure soliton unique features such as power oscillations and singularity formation. The proposed can be viewed discretization of recently obtained nonlocal Schr\"odinger equation.

In 2013, a new nonlocal symmetry reduction of the well-known AKNS (an integrable system partial differential equations, introduced by and named after Mark J. Ablowitz, David Kaup, Alan C. Newell et al. (1974)) scattering problem was found. It shown to give rise PT symmetric Hamiltonian nonlinear Schrödinger (NLS) equation. Subsequently, inverse transform constructed for case rapidly decaying initial data family spatially localized, time periodic one-soliton solutions this paper, NLS equation...