1,849 research outputs found
Integration of partially integrable equations
Most evolution equations %or wave equations are partially integrable and, in
order to explicitly integrate all possible cases, there exist several methods
of complex analysis, but none is optimal. The theory of Nevanlinna and
Wiman-Valiron on the growth of the meromorphic solutions gives predictions and
bounds, but it is not constructive and restricted to meromorphic solutions. The
Painleve' approach via the a priori singularities of the solutions gives no
bounds but it is often (not always) constructive. It seems that an adequate
combination of the two methods could yield much more output in terms of
explicit (i.e. closed form) analytic solutions. We review this question, mainly
taking as an example the chaotic equation of Kuramoto and Sivashinsky nu u''' +
b u'' + mu u' + u^2/2 +A=0, nu nonzero, with nu,b,mu,A constants.Comment: 12 p, WASCOM XIII (Acireale, 19-25 June 2005
The Painlev\'e methods
This short review is an introduction to a great variety of methods, the
collection of which is called the Painlev\'e analysis, intended at producing
all kinds of exact (as opposed to perturbative) results on nonlinear equations,
whether ordinary, partial, or discrete.Comment: LaTex 2e, subject index, Nonlinear integrable systems: classical and
quantum, ed. A. Kundu, Special issue, Proceedings of Indian Science Academy,
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