Integrability of the Perturbed KdV Equation for Convecting Fluids: Symmetry Analysis and Solutions
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As an example of how to deal with nonintegrable systems, the nonlinear partial differential equation which describes the evolution of long surface waves in a convecting fluid ut + (uxxx + 6uux) + 5uux + (uxxx + 6uux)x = 0, is fully analyzed, including symmetries (nonclassical and contact transformatons), similarity reductions and the application of the ARS algorithm to the reductions. As a result of the calculations, the Galilean invariance of the equation is shown and all the possible solutions arising from the related ODE through these methods are obtained and classified in terms of the physical parameters. 0. Introduction Integrable systems are rare in Nature. In instead one encounters often dynamical systems described by Non Linear Partial Differential Equations (NLPDE) which in spite of its wide range of application to physical problems are unfortunately of a nonintegral type. However the definition of integrability may be given (and we have used in this paper a very precise meaning for it) and the interest still lies in dealing with such nonintegrable or almost integrable, or partially integrable Partial Differential Equations (PDE), whose particular exact solutions in the case they exist could be of paramount importance in describing such different physical processes as multilayer fluid dynamics, massive transport information through doped optical fibres, gravitycapillarity microwaves, low noise detectors based on nonclassical states of light and about one hundred more physical and even straight technological applications. This paper is a theoretical attempt in the direction of devising algorithmic procedures dealing with NLPDE which we know to be integrable from the outset. Actually we show in the first part of the paper the importance of the equation in the field of two layer fluid dynamics, but we also show how none of the known procedures based upon Painlevé Tests, Lie Classical Symmetries, Non Classical Blumen and Cole Symmetries and Contact Symmetries gives any clue of how the Equation can be treated to yield some information on the exact solutions that are known experimentally to exist. Then we turn to more advanced and still algorithmic methods (with special attention to the Singular Manifold Method) that are able to open different ways to extract information on the exact solutions of this NLPDE and at the same time can be applied to a wide range of other non linear problems. Copyright c 1996 by Mathematical Ukraina Publisher. All rights of reproduction in any form reserved. 2 J.M. CERVER´O and O. ZURR´ON The paper is divided as follows. First we derive the Equation from first principles mainly based upon the NavierStokes equation for fluids with the Rayleigh number above its critical value. In section two the problem of symmetries is analyzed in its various versions. Next we entirely devote section three to Painlevé Analysis and Similarity reductions. It is in section four where we deal with the socalled Conditional Painlevé Property and in section five where we find through the previous analysis a rich class of solutions that are then classified according to the value of the constant parameters. We close with future prospects for further work in this direction.
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TY - JOUR AU - J.M. Cerveró AU - O. Zurrón PY - 1996 DA - 1996/05/01 TI - Integrability of the Perturbed KdV Equation for Convecting Fluids: Symmetry Analysis and Solutions JO - Journal of Nonlinear Mathematical Physics SP - 1 EP - 23 VL - 3 IS - 1-2 SN - 1776-0852 UR - https://doi.org/10.2991/jnmp.1996.3.1-2.1 DO - 10.2991/jnmp.1996.3.1-2.1 ID - Cerveró1996 ER -