摘要:

The hierarchies of evolution equations associated with the spectral operators∂x∂y− R∂y− Qand∂x∂y− Qin the plane are considered. In both cases a recursion operator Ф12, which is nonlocal and generates the hierarchy, is obtained. It is shown that only in the first case does the recursion operator satisfy the canonical geometrical scheme in 2 + 1 dimensions proposed by Fokas and Santini. The general procedure proposed allows one to derive, at the same time, the evolution equations associated with a given linear spectral problem and their Backlund transformations (if they exist), with no need to verify by long and tedious computations the algebraic properties of Ф12. Two equations in the first hierarchy can be considered as two different integrable generalizations in the plane of the dispersive long wave equation. All equations in this hierarchy are shown to be both a dimensional reduction of bi‐Hamiltoniann×nmatrix evolution equations in multidimensions and a generalization in the plane of bi‐Hamiltoniann×nmatrix evolutio

ISSN:0022-2526    

DOI:10.1002/sapm19887811

年代:1988

数据来源: WILEY

摘要:

An alternative approach to the classical ballot problem leads to some classes of special functions.

ISSN:0022-2526    

DOI:10.1002/sapm198878121

年代:1988

数据来源: WILEY

摘要:

The time evolution of a small disturbance on a piecewise linear mean flow, approximating a parabolic profile, is calculated using Fourier transform methods. The solution is found to consist of two parts: one dispersive, incorporating the spreading of waves; one convective, characterized by a convection of the disturbance with the local mean velocity. Two dispersive modes are found: one symmetric with respect to the channel center line and one antisymmetric. The dispersivity of the symmetric mode is in fair agreement with the symmetric mode obtained for inviscid parabolic flow, whereas the antisymmetric mode is misrepresented. One of the parts of the solution to the horizontal velocities is found to be purely three‐dimensional. This results from fluid elements retaining part of their horizontal momentum as they are lifted up by the time integrated effect of the vertical velocity. Calculations of the development of a particular disturbance modeling two vortex pairs are also made. The results show that the dispersive part, although decaying, is largest for the vertical velocity. For the horizontal velocity the three‐dimensional lift‐up effect provides the largest amplitudes. This part does not show any sign of decay, in agreement with earlier analysis by Gustavsson [8] and Landahl [16]. This last effect partly explains the sensitivity to three‐dimensional disturbances seen in transition experiments and calculations. Comparison of the solution to a full numerical simulation using the Navier‐Stokes equations shows good agreement for sh

ISSN:0022-2526    

DOI:10.1002/sapm198878131

年代:1988

数据来源: WILEY

摘要:

A theoretical study is made of the free‐surface flow induced by a wavemaker, performing torsional oscillations about a vertical axis, in a shallow rectangular channel near a cut‐off frequency. Exactly at cut‐off, linearized water‐wave theory predicts a temporally unbounded response due to a resonance phenomenon. It is shown, through a perturbation analysis using characteristic variables, that the nonlinear response is governed by a forced Kadomtsev—Petviashvili (KP) equation with periodic boundary conditions across the channel. This nonlinear initial‐boundary‐value problem is investigated analytically and numerically. When surface‐tension effects are negligible, the nonlinear response reaches a steady state and exhibits jump phenomena. On the other hand, in the high‐surface‐tension regime, no steady state is obtained. These results are discussed in connection with similar forced wave phenomena studied previously in a deepwater channel and related lab

ISSN:0022-2526    

DOI:10.1002/sapm198878157

年代:1988

数据来源: WILEY

摘要:

Nonlinear dispersive oscillatory waves are analyzed for Korteweg‐deVries type partial differential equations with slowly varying coefficients and arbitrary small perturbations. Spatial and temporal evolution of the amplitude parameters are determined in the usual way by the possible dissipation of the wave actions for both momentum and energy. For dissipative perturbations, both wave actions are shown to be valid to a higher order. Thus, the first variation of the wave action equations is used to derive equations for the slow modulations of the phase shift. It is shown that the phase shift satisfies a set of two coupled linear equation

ISSN:0022-2526    

DOI:10.1002/sapm198878173

年代:1988

数据来源: WILEY