The thermodynamic limit of the Whitham equations
نویسنده
چکیده
The infinite-genus limit of the KdV–Whitham equations is derived. The limit involves special scaling for the associated spectral surface such that the integrated density of states remains finite as N →∞ (the thermodynamic type limit). The limiting integro-differential system describes slow evolution of the density of states and can be regarded as the kinetic equation for soliton gas. 2003 Elsevier Science B.V. All rights reserved.
منابع مشابه
ar X iv : 0 81 0 . 24 27 v 3 [ m at h - ph ] 2 1 A pr 2 00 9 The multicomponent 2 D Toda hierarchy : dispersionless limit Manuel Mañas and Luis Mart́ınez
The factorization problem of the multi-component 2D Toda hierarchy is used to analyze the dispersionless limit of this hierarchy. A dispersive version of the Whitham hierarchy defined in terms of scalar Lax and Orlov–Schulman operators is introduced and the corresponding additional symmetries and string equations are discussed. Then, it is shown how KP and Toda pictures of the dispersionless Wh...
متن کاملModified F-Expansion Method Applied to Coupled System of Equation
A modified F-expansion method to find the exact traveling wave solutions of two-component nonlinear partial differential equations (NLPDEs) is discussed. We use this method to construct many new solutions to the nonlinear Whitham-Broer-Kaup system (1+1)-dimensional. The solutions obtained include Jacobi elliptic periodic wave solutions which exactly degenerate to the soliton solutions, triangu...
متن کاملNumerical Solution of the Small Dispersion Limit of Korteweg De Vries and Whitham Equations
The Cauchy problem for the Korteweg de Vries (KdV) equation with small dispersion of order ǫ, ǫ ≪ 1, is characterized by the appearance of a zone of rapid modulated oscillations of wave-length of order ǫ. These oscillations are approximately described by the elliptic solution of KdV where the amplitude, wave-number and frequency are not constant but evolve according to the Whitham equations. In...
متن کامل0 40 40 05 v 2 2 9 A pr 2 00 4 Whitham hierarchy in growth problems ∗
We discuss the recently established equivalence between the Laplacian growth in the limit of zero surface tension and the universal Whitham hierarchy known in soliton theory. This equivalence allows one to distinguish a class of exact solutions to the Laplacian growth problem in the multiply-connected case. These solutions corerespond to finite-dimensional reductions of the Whitham hierarchy re...
متن کامل2 00 4 Whitham hierarchy in growth problems ∗
We discuss the recently established equivalence between the Laplacian growth in the limit of zero surface tension and the universal Whitham hierarchy known in soliton theory. This equivalence allows one to distinguish a class of exact solutions to the Laplacian growth problem in the multiply-connected case. These solutions corerespond to finite-dimensional reductions of the Whitham hierarchy re...
متن کامل