A Priori Estimates for Solutions of a Nonlinear Dispersive Equation
نویسندگان
چکیده
In this work we obtain some a priori estimates for a higher order Schrödinger equation and in particular we obtain some a priori estimates for the modified Korteweg-de Vries equation.
منابع مشابه
A Singular Limit Problem for Conservation Laws Related to the Camassa-holm Shallow Water Equation
We consider a shallow water equation of Camassa-Holm type, containing nonlinear dispersive effects as well as fourth order dissipative effects. We prove that as the diffusion and dispersion parameters tend to zero, with a condition on the relative balance between these two parameters, smooth solutions of the shallow water equation converge to discontinuous solutions of a scalar conservation law...
متن کاملA Dispersive Estimate for the Linear Wave Equation with an Electromagnetic Potential
We consider radial solutions to the Cauchy problem for the linear wave equation with a small short–range electromagnetic potential (the“square version”of the massless Dirac equation with a potential) and zero initial data. We prove two a priori estimates that imply, in particular, a dispersive estimate.
متن کاملLocal Smoothing Effects for the Water-wave Problem with Surface Tension Hans Christianson, Vera Mikyoung Hur, and Gigliola Staffilani
Dispersive characters are studied for waves on the one-dimensional free surface of an infinitely deep perfect fluid under the influence of gravity and surface tension. The hydrodynamic problem for surface water-waves is discussed with emphasis on the effects of surface tension. A new formulation is developed as a second-order in time quasilinear dispersive equation for a dynamic variable define...
متن کاملOptimal order finite element approximation for a hyperbolic integro-differential equation
Semidiscrete finite element approximation of a hyperbolic type integro-differential equation is studied. The model problem is treated as the wave equation which is perturbed with a memory term. Stability estimates are obtained for a slightly more general problem. These, based on energy method, are used to prove optimal order a priori error estimates.
متن کاملFully Discrete Schemes for the Schrödinger Equation. Dispersive Properties
We consider fully discrete schemes for the one dimensional linear Schrödinger equation and analyze whether the classical dispersive properties of the continuous model are presented in these approximations. In particular Strichartz estimates and the local smoothing of the numerical solutions are analyzed. Using a backward Euler approximation of the linear semigroup we introduce a convergent sche...
متن کامل