Wave Propagation in Dissipative or Dispersive Non-linear Media

A singular perturbation method is developed to investigate onedimensional weak nonlinear waves in dissipative or dispersivemedia. Utilizing this method a boundary value problem for asystem of partial differential equations characterizing wavepropagation in homogeneous dissipative or dispersive media isstudied. In order to obtain a first-order uniformly valid solution,the problem is reduced to an initial value problem for scalarnon-linear partial differential equation. Some special casesarising from the structure of coefficient matrices are examinedand the method is extended to these cases. As an applicationof the perturbation method, various problems of wave propagationin a finite linear viscoelastic half-space are studied.     

Applications of Slowly Varying Nonlinear Dispersive Wave Theories

Previous theories for the slow dispersion of singly periodic and multiply periodic nonlinear wavetrains are applied to various examples. Stability of the solution with respect to large scale variations and possible shocks in the wave properties (frequency, wavenumber, and amplitude) are discussed. The general approach is also used to analyze certain aperiodic waves.     

Branching Near Plane Waves in Perturbed Dispersive Systems

A Poincaré-Lindstedt type technique for partial differential equations is used to study branching phenomena in perturbed dispersive systems arising in hydrodynamic stability theory. Multi-periodic waves with two frequencies which branch from a family of neutrally stable nonlinear periodic plane waves are constructed, the second frequency as a power series expansion in ε. The branching is compared with that of the unperturbed equations described in an earlier paper for the purpose of understanding how higher order perturbation terms effect the properties of the lower order amplitude equations. We find that in general the perturbation terms alter the leading order frequency shifts, thus changing the bifurcation from pitchfork to transverse type. The method is used to study the perturbed nonlinear Schrödinger equation and the perturbed MKdV equation.     

色散长波方程族的换位表示

本文据文［１］的思路，得到了色散长波方程族的换位表示，并讨论了驻定色散长波系统．     

Nonlinear Dispersive Systems: Theory and Examples

Some recent results in the asymptotic theory of differential equations are applied to certain third-order scalar and vector boundary-value problems that model various nonlinear physicochemical and dispersive wave phenomena. The key to our approach is the reduction of the third-order problems to asymptotically equivalent second-order ones which are more amenable to analysis. Many examples are discussed, including the problem of solitary-wave solutions of generalized Korteweg-de Vries equations and coupled systems of such equations.     

一类双色散波方程的Cauchy问题

This paper concerns with the Cauchy problem for the nonlinear double dispersive wave equation.By the priori estimates and the method in [9],It proves that the Cauchy problem admits a unique global classical solution.And by the concave method,we give sufficient conditions on the blowup of the global solution for the Cauchy problem.     

Approximate Methods for Obtaining Multi-Phase Modes in Nonlinear Dispersive Wave Problems

Perturbation and numerical methods are used to find singly periodic and multiply periodic solutions to equations which arise in the study of nonlinear wave propagation. Approximation techniques such as these are essential in those cases where exact representations of the modes are not feasible.     

Phase Resonances of the NLS Rogue Wave Recurrence in the Quasisymmetric Case

Based on experimental observations of the recurrence of anomalous waves in water and nonlinear optics, we investigate the theory of anomalous waves for initial data almost satisfying the symmetry conditions in the experiment. We also derive useful formulas, in particular, describing the phase resonance in the recurrence, which can be compared with both the currently available experimental data and the experimental data to be obtained in the near future.     

Numerical Solution of Some Nonlocal, Nonlinear Dispersive Wave Equations

Summary. We use a spectral method to solve numerically two nonlocal, nonlinear, dispersive, integrable wave equations, the Benjamin-Ono and the Intermediate Long Wave equations. The proposed numerical method is able to capture well the dynamics of the solutions; we use it to investigate the behaviour of solitary wave solutions of the equations with special attention to those, among the properties usually connected with integrability, for which there is at present no analytic proof. Thus we study in particular the resolution property of arbitrary initial profiles into sequences of solitary waves for both equations and clean interaction of Benjamin-Ono solitary waves. We also verify numerically that the behaviour of the solution of the Intermediate Long Wave equation as the model parameter tends to the infinite depth limit is the one predicted by the theory. Received October 28, 1997; revised February 11, 1999; accepted April 7, 1999     

Periodic Oscillations and Waves in Nonlinear Weakly Coupled Dispersive Systems

Bifurcations of periodic solutions in autonomous nonlinear systems of weakly coupled equations are studied. A comparative analysis is carried out between the mechanisms of Lyapunov–Schmidt reduction of bifurcation equations for solutions close to harmonic oscillations and cnoidal waves. Sufficient conditions for the branching of orbits of solutions are formulated in terms of the Pontryagin functional depending on perturbing terms.     

Contributions to the Theory of Waves in Non-linear Dispersive Systems

This paper makes contributions to the general theory of wavepropagation in conservative systems under conditions when theproportional change in amplitude or wavenumber over a distanceof one wavelength is very small. For linear systems, such propagationis governed by the well-known theory of group velocity; thereis "frequency dispersion", in the sense that energy in componentsof different frequency is propagated at different group velocities.For non-linear systems without frequency dispersion, e.g. acousticsystems, a different, but also well-known, modification of thewaveform occurs. It may be called "amplitude dispersion", inthat different values of an amplitude variable like the pressureare propagated at different speeds. A much more general theory of non-linear systems, where frequencydispersion and amplitude dispersion would be expected to bein competition, has been given by Whitham (1965b). Energy doesnot play a key role in the theory, because it is easily transferredbetween components of different frequencies. The fundamentalequation follows from Hamilton's principle in an averaged form. In examples given by Whitham, changes in, for example, wavenumber(or amplitude) are propagated at two different velocities, becausethe fundamental equation is hyperbolic. However, in the limitingcase of infinitesimal amplitude, the equation is parabolic andonly one velocity of propagation (the group velocity) occurs.Thus, Whitham showed that non-linearity can "split" the groupvelocity. This paper is concerned with the inference of detailed conclusionsfrom Whitham's theory, to enable comparisons with experimentthat will show the range of applicability of the theory. Itattempts to obtain these in the simplest case, namely, thatof one-dimensional propagation when Whitham's "pseudo-frequencies"are absent. If the relationship between frequency and wavenumber k forinfinitesimal amplitude is = f(k), then for finite amplitudethe equation is shown to be hyperbolic or elliptic respectively,according as [–f(k)]f^*(k) takes positive or negative values.For gravity waves on deep water this product is negative andthese, it is inferred, may be good for comparison of theorywith experiment in the elliptic case. A new non-linear non-perturbationaltheory of waves under the combined action of gravity and surfacetension is used to indicate that waves at 9.6 c/s on mercurymay be suitable for comparison with experiment in the hyperboliccase. When non-linear effects are only moderate, approximate transformationsof Whitham's equation to the axisymmetric Laplace and wave equationsrespectively, in the elliptic and hyperbolic cases, are usedto obtain particular solutions for comparison with experiment.A feature of these solutions is the appearance of discontinuitiesin wavelength. For example, when a wavemaker creates gravity waves of fixedfrequency whose amplitude first increases and then decreases,the theory predicts that the length of waves in the group decreasesahead of the point of maximum amplitude and increases behindit. This produces in turn a concentration of energy towardsthe centre of the group, which continues during the whole periodbefore a discontinuity in wavelength actually forms. This solutionin the elliptic case is obtained with the aid of the theoryof imaginary characteristics.     

Shallow Water Wave Systems

In this article we study various systems that represent the shallow water wave equation
v_xxt + αvv_t − βv_x∂_x^-1 ( v_t ) − v_t − v_x = 0,
where (∂ _x ⁻¹ f )( x )=∫ _x ^∞ f ( y ) d y , and α and β are arbitrary, nonzero, constants. The classical method of Lie, the nonclassical method of Bluman and Cole [ J. Math. Mech. 18:1025 (1969)], and the direct method of Clarkson and Kruskal [ J. Math. Phys. 30:2201 (1989)] are each applied to these systems to obtain their symmetry reductions. It is shown that for both the nonclassical and direct methods unusual phenomena can occur, which leads us to question the relationship between these methods for systems of equations. In particular an example is exhibited in which the direct method obtains a reduction that the nonclassical method does not.     

Elastic Wave Velocities in Two-component Systems

The problem is formulated for the scattering of long wavelengthplane elastic waves by a single homogeneous obstacle in an infiniteelastic medium. The problem differs from earlier studies inthat discrete differences in the physical properties are permittedto exist at the boundary. Earlier treatments of the scatteringproblem in which the Green's function for the scatterer is asimple operation on the incident field are inadequate to accountfor the refraction of the field at the boundary. For the caseof scattering of long waves by a spherical obstacle the scatteredfields are shown to involve the elastic constants in identicallythe same way as does the static field for the same geometryin the presence of a uniform static field. As a special casea new solution is given for the static problem of the inhomogeneousinclusion. A wave equation is derived for the "average" fielddue to multiple scattering by a statistical distribution ofspheres. The macroscopic wave parameters for the long wavelengthapproximation are obtained as a weighted contribution of theproperties of the two components.     

Exact Solutions for Large Amplitude Waves in Dispersive and Dissipative Systems

This paper describes some applications of a transformation which takes a system of two, first order, quasilinear, partial differential equations in two dependent and two independent variables into a similar system. Typically, the equations govern wave propagation in dispersive and dissipative systems. It is shown that certain nonlinear equations which are of current practical interest can be transformed into linear equations.     

On the Stability of Equilibria in Two-Degrees-of- Freedom Hamiltonian Systems Under Resonances

We consider the problem of stability of equilibrium points in Hamiltonian systems of two degrees of freedom under resonances. Determining the stability or instability is based on a geometrical criterion based on how two surfaces, related with the normal form, intersect one another. The equivalence of this criterion with a result of Cabral and Meyer is proved. With this geometrical procedure, the hypothesis may be extended to more general cases.     

一般变系数耗散长水波方程的精确解

By using the homogeneous balance principle(HBP),we derive a B■cklund trans- formation(BT)to the generalized dispersive long wave equation with variable coefficients. Based on the BT,we give many kinds of the exact solutions of the equatioh,such as,single solitary solutions,multi-soliton solutions and generalized exact solutions.     

Nonlinear Wave Interactions in Nonlinear Nonintegrable Systems

The main goal of this article is to understand the qualitative appearances of regular arrays of pulses that come up in nonintegrable systems in a variety of contexts, particularly in fluid dynamics. It is shown that even nonintegrable systems have a kind of particle dynamics made up of solitary waves. But the interaction of these solitary waves is not absolutely “clean” as in the case of the KdV and other integrable equations.     

关于(2+1)-维色散的长波方程求解的一个注记

在文献[3]的基础上，根据一些简单方程的特征，导出了(2十1)—维色散的长波方程的新的精确解，其中包含了已有文献中的孤子解，多孤子解等．