Nonlinear schrodinger equations and the universal description. In both cases, the interaction region is described by a modulated, quasiperiodic twophase solution of the nonlinear schrodinger equation. Strichartz estimates, bilinear estimates, modulation and adapted function spaces, with an application to the generalized kortewegde vries equation and the kadomtsevpetviashvili equation. Nonlinear dispersive wave systems nonlinear dispersive wave.
We show that a dispersive wave equation is obtained from a nonlocal elastic constitutive law, based on a mixture of a local and a nonlocal. An equation is globally wellposed gwp if one can take t arbitrarily large. Lectures on linear and nonlinear dispersive waves draft in. Suppose that our dispersive medium extends from, where it interfaces with a vacuum, to. Wave propagation in dispersive media let us investigate the propagation of electromagnetic radiation through a general dispersive medium by studying a simple onedimensional problem. The dispersive 1d wave equation physical audio signal. Comparison of finite difference schemes for the wave. In all of the systems weve studied longitudinal springmass, transverse string, longitudinal sound, we ended up with a wave equation of. Dispersive estimates in rn n2 for the schrodinger and the. Chapter 4 schroedinger equation einsteins relation between particle energy and frequency eq.
Both examples can easily be reproduced in a container of water one that is. We establish the local wellposedness for a new nonlinearly dispersive wave equation and we show that the equation has solutions that exist for indefinite times as well as solutions which blowup. Wellposedness for a higherorder, nonlinear, dispersive equation on a quarter plane. The table of contents for the full book pdf is as follows. In the ideal vibrating string, the only restoring force for transverse displacement comes from the string tension g. We obtain a multiscale wave packet representation for the fundamental solution of the wave equation whose coe. It arises in fields like acoustics, electromagnetics, and fluid dynamics. Boussinesq equation longitudinal waves for elasticity. Non dispersive wave equation a second order linear partial differential equation. M boiti et al 1987 inverse problems 3 371 view the article online for updates and enhancements. We conclude that the most general solution to the wave equation, is a superposition of two wave disturbances of arbitrary shapes that propagate in opposite directions, at the fixed speed, without changing shape.
The first part of the book provides an introduction to key tools and techniques in dispersive equations. Dispersive and strichartz estimates for the wave equation inside cylindrical convex domains by len meas in this work, we establish local in time dispersive estimates and its application to strichartz estimates for solutions of the model case dirichlet wave equation inside cylindrical convex domains. Blochwave homogenization on large time scales and dispersive e. On blowup criteria for a class of nonlinear dispersive wave.
Oct 10, 2017 immediate online access to all issues from 2019. Let us recall that dispersive estimates are key ingredients to prove strichartz estimates. Simon moulin nantes, lmjl dispersive estimates 11 avril 2008 orl. Chapter 4 the wave equation another classical example of a hyperbolic pde is a wave equation. Pdf a dispersive wave equation using nonlocal elasticity. A dispersive wave equation using nonlocal elasticity. Phys 201 disperion 1 a dispersive wave equation youtube. In the ideal vibrating string, the only restoring force for transverse displacement comes from the string tension c. Dispersive equations and nonlinear waves springerlink. Certain dispersive systems support sinusoidal waves only if the frequency is above or below a certain cuto. Wellposedness for nonlinear dispersive and wave equations.
On dispersive equations and their importance in mathematics. Phys 201 disperion 1 a dispersive wave equation physierge. In making the animation, 100 sinusoidal waves are superposed for a dispersion relation, the dispersion relation describes the ion acoustic wave in a plasma and also approximately shallow water waves. Besides frequency dispersion, water waves also exhibit amplitude dispersion. A residue representation for integrable equations in multidimensions.
Pdf dispersive and strichartz estimates for the wave. Dispersion relations, stability and linearization 1. This is a nonlinear effect, by which waves of larger amplitude have a different phase speed from smallamplitude waves. Eventually the two merge into a single dispersive shock wave. Blowup phenomena for a new periodic nonlinearly dispersive. When the dispersive hydrodynamics admit the above properties, we say that they are of kdv type.
The idea of dispersion is introduced, and a dispersive wave equation is derived. The paper explores comparably low dispersive scheme with among the finite difference schemes. Dispersive and strichartz estimates for the wave equation. We then present a precise blowup scenario and several blowup results of strong solutions to the equation. These four equations are of course only a very small sample of the nonlinear dispersive equations studied in the literature, but they are reasonably representative in that they showcase many of the techniques used for more general. What is a dispersive equation the simplest possible evolution partial di. In the other case, a fast dispersive shock overtakes a slower one, giving rise to an interaction. For the wave equation utt c2uxx, we plug in a wave like solution 1 to get. Pdf for a higherdimensional integrable nonlinear dynamical system, there are. Feb 10, 2018 phys 201 disperion 1 a dispersive wave equation physierge. An equation connecting the vibration frequency with the wave vector of a planar wave. Nonlinear dispersive wave equations for microstructured solids.
If the medium is dispersive different frequencies travel at. Nonlinear dispersive wave systems nonlinear dispersive. Higher order finite difference discretization for the wave equation the two dimensional version of the wave equation with velocity and acoustic pressure v in homogeneous mu edia can be written as 2 22 2 2 22, u uu v t xy. Dispersive estimates for wave equations with rough coefficients daniel tataru and danandrei geba abstract. The wave equation is a secondorder linear hyperbolic pde that describesthe propagation of a variety of waves, such as sound or water waves. This is the socalled dispersion relation for the above wave equation. Non dispersive wave equation a second order linear partial differential equation provided. A formulation of a boundaryinitialvalue problem for the kortewegde vries equation.
Local and global analysis of nonlinear dispersive and wave. The concepts and methods covered include wave dispersion, asymptotic analysis, perturbation theory, the method of multiple scales, deep and shallow water waves, nonlinear optics including fiber optic communications, modelocked lasers and dispersionmanaged wave phenomena. Thus we recognize that v represents the wave velocity. Demonstrate that both standing and traveling waves satisfy this equation hw example. Singlemode dispersive waves and soliton microcomb dynamics. Animation 1 below shows propagation of dispersive wave packet and animation 2 nondispersive wavepacket. September 2, 20 abstract we investigate second order linear wave equations in periodic media, aiming at the derivation of e. Dispersive estimates in rn n2 for the schrodinger and. The wave equation is an important secondorder linear partial differential equation for the description of wavesas they occur in classical physicssuch as mechanical waves e.
Certain dispersive systems support sinusoidal waves only if the frequency is above or below a. This search for an equation describing matter waves was carried out by erwin schroedinger. Wellposedness for regularized nonlinear dispersive wave. Blochwave homogenization on large time scales and dispersive. Demonstrate that both standing and traveling waves satisfy this equation hw. These basic requirements are quite generic and are satisfied by many dispersive hydrodynamic equations arising in applications. The main advantage of these models is their simplicity on one hand, and the ability of providing good quantitative predictions on the other hand. The ambition of these pages is to try to summarize the state of the art concerning the local and global wellposedness of common dispersive and wave equations, particularly with regard to the question of low regularity data. Spectral transform for a two spatial dimension extension of. On the cauchy problem for a nonlinearly dispersive wave equation. Consistent with geometrical optics, its magnitude is constrained to be proportional to the refractive index n 2. Chapter 4 schroedinger equation mit opencourseware. If c c g which, as we shall see, is the case for deep water waves, new wave crests appear at the rear of the wave packet, move forward.
The wave evolves according to the exponential law the wave evolves according to the exponential law the dispersion equation is deduced from the equations describing the process under observation, and defines the dispersion of the wave see, for example, the. A plot of the recoil shift determined this way versus the dispersivewave power is given as the inset in fig. In making the animation, 100 sinusoidal waves are superposed for a dispersion relation, the dispersion relation describes the ion acoustic wave in. The boussinesq equation the obtained dispersive wave equation can be reduced to the boussinesq equation under following. In this work, we establish local in time dispersive estimates and its application to strichartz estimates for solutions of the model case dirichlet wave equation inside cylindrical convex domains. In the presence of absorption, the wave will diminish in size as it move. Related content on the spectral transform of a kortewegde vries equation in two spatial dimensions m boiti, j j p leon, m manna et al. A wave is disturbance of a continuous medium that propagates with a fixed shape at constant velocity. Numerical solution of nonlinear wave equations in strati. Comparison of finite difference schemes for the wave equation. Mar 23, 2017 a plot of the recoil shift determined this way versus the dispersive wave power is given as the inset in fig. Dissipation, dispersion, and group velocity people.847 1325 143 1062 1208 416 709 530 1473 1203 843 1138 80 662 1304 1181 873 10 574 632 445 239 984 402 444 841 315 128 585 98 1406 1299 254 536 894