Compatibility conditions to allow some large amplitude WKB analysis for Burger's type systems

PHYSICA D-NONLINEAR PHENOMENA

卷:237

Article; Proceedings Paper

Cheverry, C.¹  Houbad, M.¹ 

[1] Univ Rennes 1, CNRS, UMR 6625, IRMAR, F-35042 Rennes, France

关键词: Burger and Euler equations;    nonlinear geometric optics;   

DOI  :  10.1016/j.physd.2008.03.022

来源: Elsevier

【 摘 要 】

In this article, we discuss the problem of finding large amplitude asymptotic expansions for monophase oscillating solutions of the following multidimensional (d > 1) Burger's type system: (lozenge)partial derivative(t)u+(V o u . del(x))u = 0, u is an element of R-d, (t , x ) is an element of R x R-d, V is an element of C-1 (R-d; R-d ) More precisely, we are concerned with families {u(epsilon)}(epsilon is an element of vertical bar 0,1 vertical bar) made of solutions to (lozenge) and having a development of the form u(epsilon)(t , x) = H (t, x, Phi(t. x)/epsilon) + O(epsilon) where the function theta (bar right arrow) H(t, x, theta) is periodic. In general, due to the formation of shocks, such a construction is not possible on a domain Omega which does not depend on epsilon is an element of]0,1]. In this article, we perform a detailed analysis of the restrictions to impose on the profile H and on the phase Phi in order to remedy this. Among these compatibility conditions, we can isolate some new interesting system of nonlinear partial differential equations. We explain how to solve them and we describe how the underlying structure is propagated through the evolution equation. (c) 2008 Elsevier B.V. All rights reserved.

【 授权许可】

Free