The basic content of this survey is an exposition a recently developed method constructing broad class periodic and almost-periodic solutions non-linear equations mathematical physics to which (in the rapidly decreasing case) inverse scattering problem applicable. These are such that spectrum their associated linear differential operators has finite-zone structure. set with given Jacobian variety Riemann surface, determined by structure spectrum. We give explicit solution corresponding in...

Abstract. We construct a multi-parametric family of quasi-rational solutions to the focusing NLS equation, presenting profile multiple rogue waves. These have also been used by us large smooth, real localized rational KP-I equation quite different from multi-lumps first constructed in Bordag et al. (1977). The physical relevance both equations is very large. From point view geosciences,the relevant description surface waves deep water, and occurs capillary gravitational on liquid surface,...

Our discovery of multi-rogue wave (MRW) solutions in 2010 completely changed the viewpoint on links between theory rogue waves and integrable systems, helped explain many phenomena which were never understood before. It is enough to mention famous Three Sister observed oceans, creation a regular approach studying higher Peregrine breathers, new understanding 2 + 1 dimensional via NLS-KP correspondence. This article continues study MRW NLS equation their with KP-I started previous series...

The method of finite-gap integration was created to solve the periodic KdV initial problem. Its development during last 30 years, combining spectral theory differential and difference operators with coefficients, algebraic geometry compact Riemann surfaces their Jacobians, theta functions inverse problems, had a strong impact on evolution modern mathematics theoretical physics. This article explains some principal historical points in creation this period 1973-1976, briefly comments its years.

We describe a unified structure of rogue wave and multiple solutions for all equations the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy their mixed deformed versions. The definition AKNS its versions is given in Sec. II. also consider continuous symmetries related spectral curves. This work continues summarises some our previous studies dedicated to wave-like associated with AKNS, nonlinear Schrödinger, KP hierarchies. general scheme illustrated by examples small rank n, n ⩽ 7, rational or...

CONTENTS Introduction Chapter I. Reduction of Abelian integrals and theta functions § 1. Riemann 2. genus 2 3. Normal coverings the reduction II. Multiphase (finite-zone) solutions, expressed by Jacobi functions, non-linear equations KdV-type g ≥ 4. Solutions sine-Gordon equation elliptic 5. Two-zone Lame potentials associated hyperelliptic 6. On a periodic solution problem Kovalevskaya 7. Landau-Lifschitz References

The concept of positons is introduced for the Toda lattice equation. It shown that these multiparametric oscillating and slowly decaying solutions, when inserted as potentials in finite-difference Schrodinger equation corresponding Lax pair, lead to a trivial S-matrix. resulting eigenvalues are embedded continuous spectrum this infinite Jacobi matrix. singularities connected with one-positon solution discussed compared those integrable models. special features soliton-positon interaction analysed.

The asymptotics of the τ function generating N-positon–M-soliton solution Korteweg–de Vries equation is calculated. This allows to prove that solitons do not experience any phase shift in a collision with positons. positons themselves survive mutual collisions unchanged. phenomenon called supertransparency or super-reflectionless property multipositon solutions. linear aspects this are also discussed. It demonstrated acquire two additional but always finite shifts solitons. result admits...