Search results for "Integrable system"
showing 10 items of 354 documents
Wronskian Addition Formula and Darboux-Pöschl-Teller Potentials
2013
For the famous Darboux-Pöschl-Teller equation, we present new wronskian representation both for the potential and the related eigenfunctions. The simplest application of this new formula is the explicit description of dynamics of the DPT potentials and the action of the KdV hierarchy. The key point of the proof is some evaluation formulas for special wronskian determinant.
Integrable models and degenerate horizons in two-dimensional gravity
1999
We analyse an integrable model of two-dimensional gravity which can be reduced to a pair of Liouville fields in conformal gauge. Its general solution represents a pair of ``mirror'' black holes with the same temperature. The ground state is a degenerate constant dilaton configuration similar to the Nariai solution of the Schwarzschild-de Sitter case. The existence of $\phi=const.$ solutions and their relation with the solution given by the 2D Birkhoff's theorem is then investigated in a more general context. We also point out some interesting features of the semiclassical theory of our model and the similarity with the behaviour of AdS$_2$ black holes.
Left braces and the quantum Yang-Baxter equation
2019
[EN] Braces were introduced by Rump in 2007 as a useful tool in the study of the set-theoretic solutions of the Yang¿Baxter equation. In fact, several aspects of the theory of finite left braces and their applications in the context of the Yang¿Baxter equation have been extensively investigated recently. The main aim of this paper is to introduce and study two finite brace theoretical properties associated with nilpotency, and to analyse their impact on the finite solutions of the Yang¿Baxter equation.
Breather Molecular Complexes in a Passively Mode‐Locked Fiber Laser
2021
International audience; Breathing solitons are nonlinear waves in which the energy concentrates in a localized and oscillatory fashion. Similarly to stationary solitons, breathers in dissipative systems can form stable bound states displaying molecule-like dynamics, which are frequently called breather molecules. So far, the experimental observation of optical breather molecules and the real-time detection of their dynamics are limited to diatomic molecules, that is, bound states of only two breathers. In this work, the observation of different types of breather complexes in a mode-locked fiber laser: multibreather molecules, and molecular complexes originating from the binding of two breat…
Integrability of the one dimensional Schrödinger equation
2018
We present a definition of integrability for the one dimensional Schroedinger equation, which encompasses all known integrable systems, i.e. systems for which the spectrum can be explicitly computed. For this, we introduce the class of rigid functions, built as Liouvillian functions, but containing all solutions of rigid differential operators in the sense of Katz, and a notion of natural boundary conditions. We then make a complete classification of rational integrable potentials. Many new integrable cases are found, some of them physically interesting.
Painlevé analysis and exact solutions for the coupled Burgers system
2006
We perform the Painleve test to a system of two coupled Burgers-type equations which fails to satisfy the Painleve test. In order to obtain a class of solutions, we use a slightly modified version of the test. These solutions are expressed in terms of the Airy functions. We also give the travelling wave solutions, expressed in terms of the trigonometric and hyperbolic functions.
On deformation of Poisson manifolds of hydrodynamic type
2001
We study a class of deformations of infinite-dimensional Poisson manifolds of hydrodynamic type which are of interest in the theory of Frobenius manifolds. We prove two results. First, we show that the second cohomology group of these manifolds, in the Poisson-Lichnerowicz cohomology, is ``essentially'' trivial. Then, we prove a conjecture of B. Dubrovin about the triviality of homogeneous formal deformations of the above manifolds.
Integrable systems, Frobenius manifolds and cohomological field theories
2022
In this dissertation, we study the underlying geometry of integrable systems, in particular tausymmetric bi-Hamiltonian hierarchies of evolutionary PDEs and differential-difference equations.First, we explore the close connection between the realms of integrable systems and algebraic geometry by giving a new proof of the Witten conjecture, which constructs the string taufunction of the Korteweg-de Vries hierarchy via intersection theory of the moduli spaces of stable curves with marked points. This novel proof is based on the geometry of double ramification cycles, tautological classes whose behavior under pullbacks of the forgetful and gluing maps facilitate the computation of intersection…
About Compactness of Faddeev Integral Equations for Three Charged Particles
1999
Momentum space three-body integral equations of the Faddeev type can not be used for Coulomb-like potentials, for energies above the breakup threshold. The reason is the occurrence of singularities in their kernels which destroy the compactness properties known to exist for purely short-range interactions. Using the rigorously equivalent formulation in terms of an effective-two- body theory, we prove that the nondiagonal kernels occurring therein possess on and off the energy shell only integrable singularities, provided all three particles have charges of the same sign (ie., only repulsive Coulomb interactions). In contrast, if some of the charges have opposite signs the nondiagonal kernel…
Spectral approach to the scattering map for the semi-classical defocusing Davey–Stewartson II equation
2019
International audience; The inverse scattering approach for the defocusing Davey–Stewartson II equation is given by a system of D-bar equations. We present a numerical approach to semi-classical D-bar problems for real analytic rapidly decreasing potentials. We treat the D-bar problem as a complex linear second order integral equation which is solved with discrete Fourier transforms complemented by a regularization of the singular parts by explicit analytic computation. The resulting algebraic equation is solved either by fixed point iterations or GMRES. Several examples for small values of the semi-classical parameter in the system are discussed.