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In a previous article we have proved non-existence of certain "solutions" of the cubically nonlinear Schr\"odinger equation in the general case, and presented solutions in the non-generic case. -- In the present article we describe a…
In this note, we generalize the nonlinearity-recovery result in [7] for classical cubic nonlinear Schr\"odinger equations to higher-order Schr\"odinger equations with a more general nonlinearity. More precisely, we consider a…
We consider the cubic-quintic nonlinear Schr{\"o}dinger equation in space dimension up to three. The cubic nonlinearity is thereby focusing while the quintic one is defocusing, ensuring global well-posedness of the Cauchy problem in the…
We introduce spatiotemporal optical dark X solitary waves of the (2+1)D {hyperbolic} nonlinear Schr\"odinger equation (NLSE), that rules wave propagation in a self-focusing and normally dispersive medium. These analytical solutions are…
We obtain analytic solution of the time-independent Schrodinger equation in two dimensions for a charged particle moving in the field of an electric quadrupole. The solution is written as a series in terms of special functions that support…
For an isotropic and homogeneous nonlinear left-handed materials, for which the effective medium approximation is valid, Maxwell's equations for electric and magnetic fields lead naturally, within the slowly varying envelope approximation,…
Under investigation in this work is an extended nonlinear Schr\"{o}dinger equation with nonzero boundary conditions, which can model the propagation of waves in dispersive media. Firstly, a matrix Riemann-Hilbert problem for the equation…
We prove global well-posedness for the cubic nonlinear Schr\"odinger equation with nonlinearity concentrated on a homogeneous Poisson process.
In this paper the one-dimensional nonparaxial nonlinear Schr\"odinger equation is considered. This was proposed as an alternative to the classical nonlinear Schr\"odinger equation in those situations where the assumption of paraxiality may…
We consider the inhomogeneous Mixed-boundary value problem for the cubic nonlinear Schr\"{o}dinger equations on the half line. We present sufficient conditions of initial and boundary data which ensure asymptotic behavior of small solutions…
In this paper we consider the long time behavior of solutions to the cubic nonlinear Schr\"odinger equation posed on the spatial domain $\mathbb{R}\times\mathbb{T}^{d}$, $1\leq d\leq4$. For sufficiently small, smooth, decaying data we prove…
High-precision approximate analytic expressions for energies and wave functions are found for arbitrary physical potentials. The Schr\"{o}dinger equation is cast into nonlinear Riccati equation, which is solved analytically in first…
In this paper, we use the variational method to find the normalized solutions of the quadratic coupled three wave Schrodinger equation with asymmetric coercive potential. We prove the existence of solutions for the system with 2-norm…
We consider the nonlinear Schr\"odinger equation with periodic dispersion management. We first establish global-in-time Strichartz estimates for the underlying linear equation with suitable dispersion maps. As an application, we establish a…
We consider the propagation of wave packets for a nonlinear Schr\"odinger equation, with a matrix-valued potential, in the semi-classical limit. For a matrix-valued potential, Strichartz estimates are available under long range assumptions.…
We study the presence of exact localized solutions in a quadratic-cubic nonlinear Schr\"odinger equation with inhomogeneous nonlinearities. Using a specific ansatz, we transform the nonautonomous nonlinear equation into an autonomous one,…
In previous paper we have shown that there is a special kind of nonlinear electrodynamics - Curvilinear Wave Electrodynamics (CWED), whose equations are mathematically equivalent to the equations of quantum electrodynamics. The purpose of…
We consider the nonlinear Schr{\"o}dinger equation with a harmonic potential in the presence of two combined energy-subcritical power nonlinearities. We assume that the larger power is defocusing, and the smaller power is focusing. Such a…
The initial value problem for the cubic defocusing nonlinear Schr\"odinger equation $i \partial_t u + \Delta u = |u|^2 u$ on the plane is shown to be globally well-posed for initial data in $H^s (\R^2)$ provided $s>1/2$. The proof relies…
We consider travelling periodic and quasiperiodic wave solutions of a set of coupled nonlinear Schr\"odimger equations. In fibre optics these equations can be used to model single mode fibers with strong birefringence and two-mode optical…