# Algebro-geometric approach to Nonlinear Integrable Equations

The KP hierarchy supplemented with a number of classical and non-classical reductions contains many integrable equations, such as the Boussinesq and nonlinear Schroedinger equations.

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Moreover, the hierarchy can be extended even further to a two-component hierarchy of Davey-Stewartson equations, which themselves contain other integrable equations. This development has deviated far from the original scope of the Kadomtsev-Petviashvili equation. As a result, many researchers nowadays use the KP equation without knowing where the equation originated and what "K" and "P" stand for.

For example, Figure 3 shows such a 2-soliton solution. For the KPII equation they are believed to be stable, although no formal proof exists. Moreover, more general multi-soliton solutions also exist for the KPII equation and describe phenomena of soliton resonance and web structure, as discussed below.

The KPI equation has a two-dimensional solitary wave called a lump. Such a solution is shown in Figure 1 above. The exact analytical form of the lump solution given above was found in by Manakov et al. Subsequently, various researchers obtained more general rational solutions of the KPI equation [e. These solutions were then reconciled with the framework of the IST by Villarroel et al. They found that, in general, the spectral characterization of the potential must include, in addition to the usual information about discrete and continuous spectrum, an integer-valued topological quantity that they called the "index" or winding number, defined by an appropriate two-dimensional integral involving both the solution of the KP equation and the corresponding scattering eigenfunction.

It was known since the original work by Kadomtsev and Petviashvili that, with respect to long transverse perturbations one-dimensional solitons are stable in the KPII equation and unstable in the KPI.

Evolution of transverse perturbations of arbitrary scale for KP1 was first investigated by [Zakharov ], where stabilization for large wave numbers was found. This finding helped to demonstrate that the instability results in the break-up of a one-dimensional soliton into a periodic chain of two-dimensional solitons [see Pelinovsky and Stepanyants ]. One-dimensional solitary waves interacting under certain angles display non-trivial web patterns which resemble those observed in the ocean.

### Geometric Study of a Family of Integrable Systems

A contour plot of such a solution is shown in Figure 5. Such a solution is the limiting case of an ordinary 2-soliton solution in which the asymptotic solitons satisfy the resonance condition. A resonant 2-soliton solution, which is the simplest of such solutions, is shown in Figure 2 above.

All of these resonant multi-leg solutions of the KPII equation are believed to be stable with respect to perturbations [see Biondini ], but no formal proof is available at present. A correspondence was also found to exist between these solutions and a special class of permutations Chakravarty and Kodama, Both variants of the KP equation admit a large family of exact periodic and quasi-periodic solutions.

Finite-genus solutions of the KPII equation were found to accurately reproduce the wave patterns measured in laboratory experiments of 2-dimensional shallow water Hammack et al. Figure 6 shows a genus-2 solution of the KPII equation.

## Algebro-geometric approach to nonlinear integrable equations - Eugene D. Belokolos - Google книги

Neue Artikel von diesem Autor. Neue Zitationen von diesem Autor. Neue Artikel, die mit der Forschung dieses Autors in Zusammenhang stehen. Artikel Zitiert von. Communications in Mathematical Physics 3 , , International Mathematics Research Notices 11 , , Journal fur die reine und angewandte Mathematik , , Transactions of the American Mathematical Society 2 , , Kruglikov, Dispersionless integrable systems in 3D and Einstein-Weyl geometry.

Nonlinear integrable equations of mathematical physics, electrical systems were studied using the algebro-geometric method in [3]. The DIST definition is given by the following infinite series Springer, Berlin R. A periodic two-phase algebro-geometric solution of the focusing of they appearance are hydrodynamics [3—6] and nonlinear optics geometric solutions of integrable nonlinear equations exists [38, Springer, B. Deconinck and H. Segur The KP equation with quasiperiodic initial data. Multiperiodic Fourier series solutions of integrable nonlinear wave equations are applied to the Algebro-Geometric Approach to Nonlinear Integrable Equations, F.

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Calogero, F. LeyvrazHow to embed an arbitrary Hamiltonian dynamics in an integrable Hamiltonian dynamics , Springer, Berlin, Heidelberg. For instance, for nonlinear evolution equations, so called 'finite gap' solutions have to Nonlinear Integrable Equations, Springer Series in Nonlinear Dynamics.

Belokolos, Alexander I. Matveev, Algebro-geometric approach to nonlinear integrable equations.

## Algebro-Geometric Approach in the Theory of Integrable Hydrodynamic Type Systems

ED Belokolos. Springer, , Classification of integrable equations on quad-graphs. The methods of algebro-geometric integration have been developed in the first place for solving nonlinear partial differential equations such as Korteveg-de Vries,. Sine-Gordon mechanical integrable systems. Algebro-geometric approach to nonlinear integrable equations springer series. The first two nonlinear systems in this soliton hierarchy read. Berlin, Germany: Springer.