Collapse in the nonlocal nonlinear Schrödinger equation

Integrable nonlocal nonlinear Schrödinger equation.

A new integrable nonlocal nonlinear Schrödinger equation is introduced. It possesses a Lax pair and an infinite number of conservation laws and is PT symmetric. The inverse scattering transform and scattering data with suitable symmetries are discussed. A method to find pure soliton solutions is given. An explicit breathing one soliton solution is found. Key properties are discussed and contras...

Chaoticons described by nonlocal nonlinear Schrödinger equation

It is shown that the unstable evolutions of the Hermite-Gauss-type stationary solutions for the nonlocal nonlinear Schrödinger equation with the exponential-decay response function can evolve into chaotic states. This new kind of entities are referred to as chaoticons because they exhibit not only chaotic properties (with positive Lyapunov exponents and spatial decoherence) but also soliton-lik...

Rogue waves in the two dimensional nonlocal nonlinear Schrödinger equation and nonlocal Klein-Gordon equation

In this paper, we investigate two types of nonlocal soliton equations with the parity-time (PT) symmetry, namely, a two dimensional nonlocal nonlinear Schrödinger (NLS) equation and a coupled nonlocal Klein-Gordon equation. Solitons and periodic line waves as exact solutions of these two nonlocal equations are derived by employing the Hirota's bilinear method. Like the nonlocal NLS equation, th...

Collapse of solitary excitations in the nonlinear Schrödinger equation with nonlinear damping and white noise.

The Nonlinear Schrödinger Equation on the Interval

Let q(x, t) satisfy the Dirichlet initial-boundary value problem for the nonlinear Schrödinger equation on the finite interval, 0 < x < L, with q 0 (x) = q(x, 0), g 0 (t) = q(0, t), f 0 (t) = q(L, t). Let g 1 (t) and f 1 (t) denote the unknown boundary values q x (0, t) and q x (L, t), respectively. We first show that these unknown functions can be expressed in terms of the given initial and bo...