摘要:

The characteristic feature of the so‐called Painlevé test for integrability of an ordinary (or partial) analytic differential equation, as usually carried out, is to determine whether all its solutions are single‐valued by local analysis near individual singular points of solutions. This test, interpreted flexibly, has been quite successful in spite of various evident flaws. We review the Painleve test in detail and then propose a more robust and generally more appropriate definition of integrability: a multivalued function is accepted as an integral if its possible values (at any given point in phase space) are not dense. This definition is illustrated and justified by examples, and a widely applicable method (thepoly‐Painlevé method) of testing for it is presented, based on asymptotic analysis covering several singularities simulta

ISSN:0022-2526    

DOI:10.1002/sapm199286287

年代:1992

数据来源: WILEY

摘要:

The expansion procedure introduced by Benney (1966) for weakly nonlinear, planar shallow‐water waves is used to provide an alternative derivation of the more general results of Benjamin (1966) for shallow fluid layers possessing arbitrary vertical stratification and horizontal shear. New solutions that include the effects of both shear and stratification are presented. The evolution equation for slowly varying cylindrical solitary waves traveling in a density‐stratified fluid is found using two‐timing techniques. Not surprisingly, one obtains the same coefficients for the nonlinear and dispersive terms as in the planar case. In the limit for uniform density it is shown that the free‐surface evolution equation of Miles (1978) for axisymmetric Boussinesq waves is re

ISSN:0022-2526    

DOI:10.1002/sapm1992862167

年代:1992

数据来源: WILEY