摘要:

A modulational stability analysis is presented for real, two‐phase sine‐Gordon wavetrains. Using recent results on the geometry of these real solutions, an invariant representation in terms of Abelian differentials is derived for the sine‐Gordon modulation equations. The theory thus attains the same integrable features of the previously completed KdV and sinh‐Gordon modulations. The twophase results are as follows: kink‐kink trains are stable, while the breather trains and kink‐radiation trains are unstable, to

ISSN:0022-2526    

DOI:10.1002/sapm198471291

年代:1984

数据来源: WILEY

摘要:

Under certain circumstances, solutions of the cylindrically symmetric nonlinear Schrödinger equation collapse to a singularity in a finite time. An asymptotic series for the solution near the singularity is derived here. At leading order, the central amplitude of the spike grows like[(log Δt)/Δt]1/2, where Δtis the time remaining to the appearance of the singular

ISSN:0022-2526    

DOI:10.1002/sapm1984712103

年代:1984

数据来源: WILEY

摘要:

This paper continues earlier work of the authors on a theory for spontaneous Mach‐stem formation. Shock formation in smooth solutions of the scalar integrodifferential conservation law from paper I is demonstrated through detailed numerical experiments—this completes the basic argument from paper I. The steady‐state bifurcation of planar detonation waves into “shallow‐angle” reactive Mach stem structures is analyzed. The conclusions of this analysis agree with those predicted through the time‐dependent asymptotics in paper I and provide a completely independent confirmation o

ISSN:0022-2526    

DOI:10.1002/sapm1984712117

年代:1984

数据来源: WILEY

摘要:

We present a systematic asymptotic theory for resonantly interacting weakly nonlinear hyperbolic waves in a single space variable. This theory includes as a special case the theory of nonresonant interacting waves for general hyperbolic systems developed recently by J. Hunter and J. B. Keller, when specialized to a single space variable. However, we are also able to treat the general situation when resonances occur in the hyperbolic system. Such resonances are the typical case when the hyperbolic system has at least three equations and when, for example, small‐amplitude periodic initial data are prescribed. In the important physical example of the 3 × 3 system describing compressible fluid flow in a single space variable, the resonant asymptotic theory developed by the authors yields, as limit equations, a pair of inviscid Burgers equations coupled through a linear integral operator with known kernel defined through the initial data for the entropy wave. (In the general case we give many new conditions guaranteeing nonresonance for a given hyperbolic system with prescribed initial data, as well as other new structural conditions which imply that resonance occurs.) A method for treating resonantly interacting waves in several space variables, together with applications, will be developed by the authors elsewhe

ISSN:0022-2526    

DOI:10.1002/sapm1984712149

年代:1984

数据来源: WILEY