Objective: To calculate wave response in an offshore or coastal harbor of arbitrary shape, this research develops a two-dimensional linear, inviscid, dispersive, hybrid finite element harbor resonance model using conservation of energy approach. Based on the mild-slope wave equation, the numerical model includes wave refraction, diffraction, and reflection. The model also incorporates the effec...

The wave equation describing an ultrashort, tightly focused laser pulse in vacuum is solved analytically. Plasma dispersive effects are also included. Based on exact short-pulse solutions, analytical expressions are obtained for the pulse-length evolution, the pulse centroid motion, and the group velocity. Approximate shortpulse solutions are obtained to arbitrary order in the parameter ly2pL ,...

The Benney-Luke equation (BL) is a model for the evolution of three-dimensional weakly nonlinear, long water waves of small amplitude. In this paper we propose a nearly conservative scheme for the numerical resolution of (BL). Moreover, it is known ([PQ99] and [Pau03]) that (BL) is linked to the Kadomtsev-Petviashvili equation for almost one-dimensional waves propagating in one direction. We st...

We prove dispersive estimates for solutions to the wave equation with a real-valued potential V ∈ L∞(R), n ≥ 4, satisfying V (x) = O(〈x〉−(n+1)/2−ǫ), ǫ > 0.

The notes serve as an introduction to the analysis of dispersive partial differential equations. They are organized as follows: • Part I focuses on basic theory for local and global analysis of the semilinear Schrödinger equation. • Part II concentrates on basic local and global theory for the Korteveg de Vries equation. • Part III gives a review of some recents results on a derivation of nonli...

We investigate traveling wave solutions to a class of dispersive models – the θ-equation of the form ut − utxx + uux = θuuxxx + (1− θ)uxuxx, including two integrable equations, the Camassa-Holm equation (θ = 1/3) and the Degasperis-Procesi equation(θ = 1/4) as special models. When 0 ≤ θ ≤ 1 2 , strong solutions of the θ-equation may blow-up in finite time, correspondingly, the traveling wave so...

We analyze a homogenization limit for the linear wave equation of second order. The spatial operator is assumed to be of divergence form with an oscillatory coefficient matrix aε that is periodic with characteristic length scale ε; no spatial symmetry properties are imposed. Classical homogenization theory allows to describe solutions uε well by a non-dispersive wave equation on fixed time inte...

Traveling wave solutions to a class of dispersive models, ut − utxx + uux = θuuxxx + (1− θ)uxuxx, are investigated in terms of the parameter θ, including two integrable equations, the Camassa-Holm equation, θ = 1/3, and the Degasperis-Procesi equation, θ = 1/4, as special models. It was proved in [H. Liu and Z. Yin, Contemporary Mathematics, 2011, 526, pp273–294] that when 1/2 < θ ≤ 1 smooth so...

Contents Preface ix Chapter 1. Ordinary differential equations 1 1.1. General theory 1 1.2. Gronwall's inequality 7 1.3. Bootstrap and continuity arguments 14 1.4. Noether's theorem 18 1.5. Monotonicity formulae 25 1.6. Linear and semilinear equations 28 1.7. Completely integrable systems 35 Chapter 2. Constant coefficient linear dispersive equations 41 2.1. The Fourier transform 47 2.2. Fundam...

Since space charge waves on a particle beam exhibit both dispersive and nonlinear character, solitary waves or solitons are possible. Dispersive, nonlinear wave propagation in high current beams is found to be similar to ion-acoustic waves in plasmas with an analogy between Debye screening and beam pipe shielding. Exact longitudinal solitary wave propagation is found for potentials associated w...