摘要:

In this paper our concern is with solutionsw(z;α,β) of the fourth Painlevé equation (PIV), whereαandβare arbitrary real parameters. It is known that PIV admits a variety of solution types and here we classify and characterise these. Using Backlund transformations we describe a novel method for efficiently generating new solutions of PIV from known ones. Almost all the established Bäcklund transformations involve differentiation of solutions and since all but a very few solutions of PIV are given by extremely complicated formulae, those transformations which require differentiation in this way are very awkward to implement in practice. Depending on the values of the parametersαandβ, PIV can admit solutions which may either be expressed as the ratio of two polynomials inz, or can be related to the complementary error or parabolic cylinder functions; in fact, all exact solutions of PIV are thought to fall in one of these three hierarchies. We show how, given a few initial solutions, it is possible to use the structures of the hierarchies to obtain many other solutions. In our approach we derive a nonlinear superposition formula which relates three solutions of PIV; the principal attraction is that the process involves only algebraic manipulations so that, in particular, no differentiation is required. We investigate the properties of our computed solutions and illustrate that they have a large number of physical appli

ISSN:0022-2526    

DOI:10.1002/sapm19959511

年代:1995

数据来源: WILEY

摘要:

In this paper we discuss a new approach to the relationship between integrability and symmetries of a nonlinear partial differential equation. The method is based heavily on ideas using both the Painlevé property and the singular manifold analysis, which is of outstanding importance in understanding the concept of integrability of a given partial differential equation. In our examples we show that the solutions of the singular manifold possess Lie point symmetries that correspond precisely to the so‐called nonclassical symmetries. We also point out the connection between the singular manifold method and the direct method of Clarkson and Kruskal. Here the singular manifold is a function of its reduced variable. Although the Painlevé property plays an essential role in our approach, our method also holds for equations exhibiting only the conditional Painlevé property. We offer six full examples of how our method works for the six equations, which we believe cover all possible c

ISSN:0022-2526    

DOI:10.1002/sapm199595173

年代:1995

数据来源: WILEY

摘要:

Approximate stationary solutions of the Ostrovsky equation describing long weakly nonlinear waves in a rotating liquid are constructed. These solutions may be regarded as a periodic sequence of arcs of parabolas containing Korteweg‐de Vries solitons at the junctures. Results of numerical computations of the dynamics of the approximate solutions obtained from the nonstationary Ostrovsky equation are presented. It is found that, in the presence of negative dispersion, the shape of a stationary wave is well predicted by the approximate theory, whereas the calculated wave velocity differs slightly from the theoretical value. The stationary solutions in media with positive dispersion are evidently unstable (at least for sufficiently strong rotation), and numerical computations demonstrate a complicated picture of nonstationary destructio

ISSN:0022-2526    

DOI:10.1002/sapm1995951115

年代:1995

数据来源: WILEY