Volume 1, Number 4, Fall 1993
EXACT SOLUTIONS OF NONLINEAR
THINFILM 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 solitarywave 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 solitarywave 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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