Error bounds for the (KdV)/(KP-I) and (gKdV)/(gKP-I) asymptotic regime for Nonlinear Schrödinger type Equations
نویسنده
چکیده
We consider the (KdV)/(KP-I) asymptotic regime for the Nonlinear Schrödinger Equation with a general nonlinearity. In a previous work, we have proved the convergence to the Korteweg-de Vries equation (in dimension 1) and to the Kadomtsev-Petviashvili equation (in higher dimensions) by a compactness argument. We propose a weakly transverse Boussinesq type system formally equivalent to the (KdV)/(KP-I) equation in the spirit of the work of Lannes and Saut, and then prove a comparison result with quantitative error estimates. For either suitable nonlinearities for (NLS) either a LandauLifshitz type equation, we derive a (mKdV)/(mKP-I) equation involving cubic nonlinearity. We then give a partial result justifying this asymptotic limit. Key-words: Nonlinear Schrödinger Equation, Gross-Pitaevskii Equation, Landau-Lifshitz equation, (generalized) Korteweg de Vries equation, (generalized) Kadomtsev-Petviashvili equation, weakly transverse Boussinesq system. MSC (2010): 35Q55, 35Q53.
منابع مشابه
Three Long Wave Asymptotic Regimes for the Nonlinear-schrödinger Equation
We survey some recent results related to three long wave asymptotic regimes for the Nonlinear-Schrödinger Equation: the Euler regime corresponding to the WKB method, the linear wave regime and finally the KdV/KP-I asymptotic dynamics.
متن کاملThe KdV/KP-I Limit of the Nonlinear Schrödinger Equation
Abstract We justify rigorously the convergence of the amplitude of solutions of Nonlinear-Schrödinger type Equations with non zero limit at infinity to an asymptotic regime governed by the Kortewegde Vries equation in dimension 1 and the Kadomtsev-Petviashvili I equation in dimensions 2 and more. We get two types of results. In the one-dimensional case, we prove directly by energy bounds that t...
متن کاملEstimation in Simple Step-Stress Model for the Marshall-Olkin Generalized Exponential Distribution under Type-I Censoring
This paper considers the simple step-stress model from the Marshall-Olkin generalized exponential distribution when there is time constraint on the duration of the experiment. The maximum likelihood equations for estimating the parameters assuming a cumulative exposure model with lifetimes as the distributed Marshall Olkin generalized exponential are derived. The likelihood equations do not lea...
متن کاملExact solutions for the family of third order Korteweg de-Vries equations
In this work we apply an extended hyperbolic function method to solve the nonlinear family of third order Korteweg de-Vries (KdV) equations, namely, the KdV equation, the modified KdV (mKdV) equation, the potential KdV (pKdV) equation, the generalized KdV (gKdV) equation and gKdV with two power nonlinearities equation. New exact travelling wave solutions are obtained for the KdV, mKdV and pKdV ...
متن کاملALGEBRAIC NONLINEARITY IN VOLTERRA-HAMMERSTEIN EQUATIONS
Here a posteriori error estimate for the numerical solution of nonlinear Voltena- Hammerstein equations is given. We present an error upper bound for nonlinear Voltena-Hammastein integral equations, in which the form of nonlinearity is algebraic and develop a posteriori error estimate for the recently proposed method of Brunner for these problems (the implicitly linear collocation method)...
متن کامل