Nonlinear Schrödinger equation with random Gaussian input: Distribution of inverse scattering data and eigenvalues

ṕ Estimates for the Schrödinger Equation on the Line and Inverse Scattering for the Nonlinear Schrödinger Equation with a Potential ∗

In this paper I prove a L − L estimate for the solutions of the one–dimensional Schrödinger equation with a potential in Lγ where in the generic case γ > 3/2 and in the exceptional case (i.e. when there is a half–bound state of zero energy) γ > 5/2. I use this estimate to construct the scattering operator for the nonlinear Schrödinger equation with a potential. I prove moreover, that the low–en...

Stekloff Eigenvalues in Inverse Scattering

We consider a problem in nondestructive testing in which small changes in the (possibly complex valued) refractive index n(x) of an inhomogeneous medium of compact support are to be determined from changes in measured far field data due to incident plane waves. The problem is studied by considering a modified far field operator F whose kernel is the difference of the measured far field pattern ...

Forced nonlinear Schrödinger equation with arbitrary nonlinearity.

We consider the nonlinear Schrödinger equation (NLSE) in 1+1 dimension with scalar-scalar self-interaction g(2)/κ+1(ψ*ψ)(κ+1) in the presence of the external forcing terms of the form re(-i(kx+θ))-δψ. We find new exact solutions for this problem and show that the solitary wave momentum is conserved in a moving frame where v(k)=2k. These new exact solutions reduce to the constant phase solutions...

Nonlinear Schrödinger equation with spatiotemporal perturbations.

We investigate the dynamics of solitons of the cubic nonlinear Schrödinger equation (NLSE) with the following perturbations: nonparametric spatiotemporal driving of the form f(x,t)=a exp[iK(t)x], damping, and a linear term which serves to stabilize the driven soliton. Using the time evolution of norm, momentum and energy, or, alternatively, a Lagrangian approach, we develop a collective-coordin...

Inverse scattering theory and transmission eigenvalues