Stability of Compacton Solutions of Fifth-Order Nonlinear Dispersive Equations

We consider fifth-order nonlinear dispersive $K(m,n,p)$ type equations to study the effect of nonlinear dispersion. Using simple scaling arguments we show, how, instead of the conventional solitary waves like solitons, the interaction of the nonlinear dispersion with nonlinear convection generates compactons - the compact solitary waves free of exponential tails. This interaction also generates many other solitary wave structures like cuspons, peakons, tipons etc. which are otherwise unattainable with linear dispersion. Various self similar solutions of these higher order nonlinear dispersive equations are also obtained using similarity transformations. Further, it is shown that, like the third-order nonlinear $K(m,n)$ equations, the fifth-order nonlinear dispersive equations also have the same four conserved quantities and further even any arbitrary odd order nonlinear dispersive $K(m,n,p...)$ type equations also have the same three (and most likely the four) conserved quantities. Finally, the stability of the compacton solutions for the fifth-order nonlinear dispersive equations are studied using linear stability analysis. From the results of the linear stability analysis it follows that, unlike solitons, all the allowed compacton solutions are stable, since the stability conditions are satisfied for arbitrary values of the nonlinear parameters.Comment: 20 pages, To Appear in J.Phys.A (2000), several modification

Interface Problems for Dispersive equations

The interface problem for the linear Schr\"odinger equation in one-dimensional piecewise homogeneous domains is examined by providing an explicit solution in each domain. The location of the interfaces is known and the continuity of the wave function and a jump in their derivative at the interface are the only conditions imposed. The problem of two semi-infinite domains and that of two finite-sized domains are examined in detail. The problem and the method considered here extend that of an earlier paper by Deconinck, Pelloni and Sheils (2014). The dispersive nature of the problem presents additional difficulties that are addressed here.Comment: 18 pages, 6 figures. arXiv admin note: text overlap with arXiv:1402.3007, Studies in Applied Mathematics 201

Pseudo-Hermiticity and Electromagnetic Wave Propagation in Dispersive Media

Pseudo-Hermitian operators appear in the solution of Maxwell's equations for stationary non-dispersive media with arbitrary (space-dependent) permittivity and permeability tensors. We offer an extension of the results in this direction to certain stationary dispersive media. In particular, we use the WKB approximation to derive an explicit expression for the planar time-harmonic solutions of Maxwell's equations in an inhomogeneous dispersive medium and study the combined affect of inhomogeneity and dispersion.Comment: 8 pages, to appear in Phys. Lett.

Spectral Theory of Time Dispersive and Dissipative Systems

We study linear time dispersive and dissipative systems. Very often such systems are not conservative and the standard spectral theory can not be applied. We develop a mathematically consistent framework allowing (i) to constructively determine if a given time dispersive system can be extended to a conservative one; (ii) to construct that very conservative system -- which we show is essentially unique. We illustrate the method by applying it to the spectral analysis of time dispersive dielectrics and the damped oscillator with retarded friction. In particular, we obtain a conservative extension of the Maxwell equations which is equivalent to the original Maxwell equations for a dispersive and lossy dielectric medium.Comment: LaTeX, 57 Pages, incorporated revisions corresponding with published versio

Smoothing estimates for non-dispersive equations

This paper describes an approach to global smoothing problems for non-dispersive equations based on ideas of comparison principle and canonical transformation established in authors' previous paper, where dispersive equations were treated. For operators $a(D_x)$ of order $m$ satisfying the dispersiveness condition $\nabla a(\xi)\neq0$ for $\xi\not=0$, the global smoothing estimate $$\|\langle x\rangle^{-s}|D_x|^{(m-1)/2}e^{ita(D_x)} \varphi(x)\|_{L^2(\mathbb R_t\times\mathbb R^n_x)} \leq C\|\varphi\|_{L^2(\mathbb R^n_x)} \quad {\rm(}s>1/2{\rm)}$$ is well-known, while it is also known to fail for non-dispersive operators. For the case when the dispersiveness breaks, we suggest the estimate in the form $$\|{\langle{x}\rangle^{-s}|\nabla a(D_x)|^{1/2} e^{it a(D_x)}\varphi(x)}\|_{L^2({\mathbb R_t\times\mathbb R^n_x})} \leq C\|{\varphi}\|_{L^2({\mathbb R^n_x})}\quad{\rm(}s>1/2{\rm)}$$ which is equivalent to the usual estimate in the dispersive case and is also invariant under canonical transformations for the operator $a(D_x)$. We show that this estimate and its variants do continue to hold for a variety of non-dispersive operators $a(D_x)$, where $\nabla a(\xi)$ may become zero on some set. Moreover, other types of such estimates, and the case of time-dependent equations are also discussed.Comment: 24 pages; the paper is to appear in Math. Ann. arXiv admin note: substantial text overlap with arXiv:math/061227