Cauchy Problem of Nonlinear Schrödinger Equation with Initial Data in Sobolev Space W
نویسنده
چکیده
In this paper, we consider in R the Cauchy problem for nonlinear Schrödinger equation with initial data in Sobolev space W s,p for p < 2. It is well known that this problem is ill posed. However, We show that after a linear transformation by the linear semigroup the problem becomes locally well posed in W s,p for 2n n+1 < p < 2 and s > n(1− 1 p ). Moreover, we show that in one space dimension, the problem is locally well posed in L for any 1 < p < 2. Keyword:Cauchy problem, nonlinear Schrödinger equation, locally well-posedness, scaling limit.
منابع مشابه
Cauchy problem of nonlinear Schrödinger equation with initial data in Sobolev space W s , p for p < 2
In this paper, we consider in R the Cauchy problem for nonlinear Schrödinger equation with initial data in Sobolev space W s,p for p < 2. It is well known that this problem is ill posed. However, We show that after a linear transformation by the linear semigroup the problem becomes locally well posed in W s,p for 2n n+1 < p < 2 and s > n(1− 1 p ). Moreover, we show that in one space dimension, ...
متن کاملCauchy Problem of Nonlinear Schrödinger Equation with Initial Data
In this paper, we consider in Rn the Cauchy problem for the nonlinear Schrödinger equation with initial data in the Sobolev space W s,p for p < 2. It is well known that this problem is ill posed. However, we show that after a linear transformation by the linear semigroup the problem becomes locally well posed in W s,p for 2n n+1 < p < 2 and s > n(1 − 1 p ). Moreover, we show that in one space d...
متن کاملNonuniqueness of Weak Solutions of the Nonlinear Schrödinger Equation
Generalized solutions of the Cauchy problem for the one-dimensional periodic nonlinear Schrödinger equation, with cubic or quadratic nonlinearities, are not unique. For any s < 0 there exist nonzero generalized solutions varying continuously in the Sobolev space H, with identically vanishing initial data.
متن کاملExact Solutions of Semiclassical Non-characteristic Cauchy Problems for the Sine-gordon Equation
The use of the sine-Gordon equation as a model of magnetic flux propagation in Josephson junctions motivates studying the initial-value problem for this equation in the semiclassical limit in which the dispersion parameter ε tends to zero. Assuming natural initial data having the profile of a moving −2π kink at time zero, we analytically calculate the scattering data of this completely integrab...
متن کاملA Priori Bounds and Weak Solutions for the Nonlinear Schrödinger Equation in Sobolev Spaces of Negative Order
Solutions to the Cauchy problem for the one-dimensional cubic nonlinear Schrödinger equation on the real line are studied in Sobolev spaces H, for s negative but close to 0. For smooth solutions there is an a priori upper bound for the H norm of the solution, in terms of the H norm of the datum, for arbitrarily large data, for sufficiently short time. Weak solutions are constructed for arbitrar...
متن کامل