摘要:

Using the forced Korteweg‐de Vries equation as a simple model, a perturbation procedure is presented for calculating the amplitude of short‐scale oscillatory tails induced by steady long‐wave disturbances. In the limit of weak dispersion, these tails have exponentially small amplitude that lies beyond all orders of the usual long‐wave expansion. It is demonstrated that by working in the wavenumber domain, the tail amplitude can be determined quite simply, without the need for asymptotic matching in the complex plane. The induced short‐wave tail is sensitive to the details of the long‐wave profile. The proposed technique is applicable to nonlocal solitary waves and to other problems that require the use of exponential

ISSN:0022-2526    

DOI:10.1002/sapm19959411

年代:1995

数据来源: WILEY

摘要:

Nonclassical symmetry methods are used to study the nonlinear diffusion equation with a nonlinear source. In particular, exponential and power law diffusivities are examined and we obtain mathematical forms of the source term which permit nonclassical symmetry reductions. In addition to the known source terms obtained by classical symmetry methods, we find new source terms which admit symmetry reductions. We also deduce a class of nonclassical symmetries which are valid for arbitrary diffusivity and deduce corresponding new solution types. This is an important result since previously only traveling wave solutions were known to exist for arbitrary diffusivity. A number of examples are considered and new exact solutions are constructed.

ISSN:0022-2526    

DOI:10.1002/sapm199594121

年代:1995

数据来源: WILEY

摘要:

The nonlinear critical layer theory is developed for the case where the critical point is close enough to a solid boundary so that the critical layer and viscous wall layers merge. It is found that the flow structure differs considerably from the symmetric “eat's eye” pattern obtained by Benney and Bergeron [1] and Haberman [2]. One of the new features is that higher harmonics generated by the critical layer are in some cases induced in the outer flow at the same order as the basic disturbance. As a consequence, the lowest‐order critical layer problem must be solved numerically. In the inviscid limit, on the other hand, a closed‐form solution is obtained. It has continuous vorticity and is compared with the solutions found by Bergeron [3], which contain discontinuities in vorticity across closed stre

ISSN:0022-2526    

DOI:10.1002/sapm199594141

年代:1995

数据来源: WILEY

摘要:

Exact free surface flows with shear in a compressible barotropic medium are found, extending the authors’ earlier work for the incompressible medium. The barotropic medium is of finite extent in the vertical direction, while it is infinite in the horizontal direction. The “shallow water” equations for a compressible barotropic medium, subject to boundary conditions at the free surface and at the bottom, are solved in terms of double psi‐series. Simple wave and time‐dependent solutions are found; for the former the free surface is of arbitrary shape while for the latter it is a damping traveling wave in the horizontal direction. For other types of solutions, the height of the free surface is constant either on lines of constant acceleration or on lines of constant speed. In the case of an isothermal medium, whenγ= 1, we again find simple wave and time‐depende

ISSN:0022-2526    

DOI:10.1002/sapm199594157

年代:1995

数据来源: WILEY

摘要:

In the classical theory of water waves only irrotational motions are considered. In the present paper a class of wave problems associated with inviscid rotational motions is examined and the consequences explored.

ISSN:0022-2526    

DOI:10.1002/sapm199594177

年代:1995

数据来源: WILEY