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Volume 1,     Number 4,     Fall 1993

 

EXACT SOLUTIONS OF NONLINEAR
THIN-FILM AMPLITUDE EVOLUTION
EQUATIONS VIA TRANSFORMATIONS
S. MELKONIAN

Abstract. For many (but not all) integrable nonlinear partial differential equations, an appropriate truncation of the local series expansion of the solution about a singular manifold reveals the Hirota transformation of the equation. The (a priori arbitrary) function φ, whose manifold of zeros determines the singular manifold, satisfies the Hirota equations when it is specialized so as to truncate the series.
In this article, Hirota's method is extended to provide a means of obtaining particular solutions of nonintegrable equations which admit solitary-wave solutions. Five equations which govern the nonlinear evolution of long waves on thin films are solved by this method. A detailed analysis of one equation is given, demonstrating that the generalized Hirota transformation does, as in some integrable cases, arise from appropriate truncation of the associated series, and that the particular function φ which guarantees such truncation necessarily satisfies the (generalized) Hirota equations. On the other hand, another one of the considered equations provides an example which demonstrates that, although the equation admits a transformation which reveals a solitary-wave solution, this cannot be obtained by truncation of the associated series. The derivation of the above equations within the context of long waves on thin films is outlined.

 

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