Riemann–Hilbert problem for Camassa–Holm equation with step-like initial data

Cauchy 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...

Quenching Time for a Nonlocal Diffusion Problem with Large Initial Data

The Inverse Scattering Transform for the Kdv Equation with Step-like Singular Miura Initial Profiles

We develop the inverse scattering transform for the KdV equation with real singular initial data q (x) of the form q (x) = r′ (x) + r (x), where r ∈ Lloc, r|R+ = 0. As a consequence we show that the solution q (x, t) is a meromorphic function with no real poles for any t > 0.

On the Whitham Equations for the Defocusing Nonlinear Schrodinger Equation with Step Initial Data

The behavior of solutions of the finite-genus Whitham equations for the weak dispersion limit of the defocusing nonlinear Schrödinger equation is investigated analytically and numerically for piecewiseconstant initial data. In particular, the dynamics of constant-amplitude initial conditions with one or more frequency jumps (i.e., piecewise linear phase) are considered. It is shown analytically...

Initial Value Problem for the Wave Equation

Nonhomeomorphic compact Hausdorff spaces X and Y and an isomorphism : C(X) -> C(Y) (onto) are constructed such that ||c/>|| \4>~ || = 2. Amir had asked if such a d> exists with || \\ ||cf> || < 3.