exact solutions of nonlinear thin-film amplitude …

CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 1, Number 4, Fall 1693 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 r e d the Hirota transformation of the equation. The (a priori arbitrary) function 4, whose manifold of zeros de- termines 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 <t> which guarantees such truncation nec- wily 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 soh- tion, this cannot be obtained by truncation of the associated series. The derivation of the above equations within the con- text of long waves on thin films is outlined. 1. Introduction. Hirota [13] has developed a direct method of obtaining the multisoliton solutions of integrable nonlinear partial differential equations (NLPDEs) involving dependent-variable transformations which cast such equations in homogeneous bilinear form. Solu- tions of the original NLPDEs having multisoliton form result provided that solutions of the transformed equations are assumed to have certain particular forms. If the number N of waves in the solution is greater than two, then the latter requirements give rise to systems of overdeter- mined algebraic equations in the soliton parameters; they are, however, consistent by virtue of the integrability of the NLPDEs. Received by the editors on August 28, 1991, and in revised form on March 30, 1993. Copyright 01993 Rocky Mountain Mathematics Consortium 493

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Page 1: EXACT SOLUTIONS OF NONLINEAR THIN-FILM AMPLITUDE …

CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 1, Number 4, Fall 1693

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 r e d the Hirota transformation of the equation. The (a priori arbitrary) function 4, whose manifold of zeros de- termines 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 <t> which guarantees such truncation nec-

wily 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 soh- tion, this cannot be obtained by truncation of the associated series. The derivation of the above equations within the con- text of long waves on thin films is outlined.

1. Introduction. Hirota [13] has developed a direct method of obtaining the multisoliton solutions of integrable nonlinear partial differential equations (NLPDEs) involving dependent-variable transformations which cast such equations in homogeneous bilinear form. Solu- tions of the original NLPDEs having multisoliton form result provided that solutions of the transformed equations are assumed to have certain particular forms. If the number N of waves in the solution is greater than two, then the latter requirements give rise to systems of overdeter- mined algebraic equations in the soliton parameters; they are, however, consistent by virtue of the integrability of the NLPDEs.

Received by the editors on August 28, 1991, and in revised form on March 30, 1993.

493