Related papers: Nonlinear Schr\"odinger equations with trapping po…
We analyse the structure of the exact, dark and bright soliton solutions of the driven non-linear Schr\"odinger equation. It is found that, a wide class of solutions of the higher order non-linear Schr\"odinger equation with a source can…
A new non-perturbative method of solution of the nonlinear Heisenberg equations in the finite-dimensional subspace is illustrated. The method, being a counterpart of the traditional Schrodinger picture method, is based on a finite operator…
We prove a global well-posedness result for defocusing nonlinear Schrodinger equations with time dependent potential. We then focus on time dependent harmonic potentials. This aspect is motivated by Physics (Bose--Einstein condensation),…
We propose a class of numerical methods for the nonlinear Schr\"odinger (NLS) equation that conserves mass and energy, is of arbitrarily high-order accuracy in space and time, and requires only the solution of a scalar algebraic equation…
We consider the cubic defocusing nonlinear Schr\"odinger equation in one dimension with the nonlinearity concentrated at a single point. We prove global well-posedness in the scaling-critical space $L^2(\mathbb{R})$ and scattering for all…
Periodic waves are standing wave solutions of nonlinear Schr\''odinger equations whose profile is periodic in space dimension one. We consider general nonlinearities and provide variational characterizations for the periodic wave profiles.…
Using similarity transformations we construct explicit nontrivial solutions of nonlinear Schr\"odinger equations with potentials and nonlinearities depending on time and on the spatial coordinates. We present the general theory and use it…
The method of solving of nonlinear Schr\"odinger equation is considered. Some examples of its applications are demonstrated.
We consider the one-dimensional nonlinear Schr\"odinger equation with Dirichlet boundary conditions in the fully resonant case (absence of the zero-mass term). We investigate conservation of small amplitude periodic-solutions for a large…
We study the instability of standing wave solutions for nonlinear Schr\"{o}dinger equations with a one-dimensional harmonic potential in dimension $N\ge 2$. We prove that if the nonlinearity is $L^2$-critical or supercritical in dimension…
We are concerned with the existence of solutions to the following nonlinear Schr\"odinger system in $\mathbb{R}^3$: \begin{equation*} \left\{ \begin{aligned} -\Delta u_1 + (x_1^2+x_2^2)u_1&= \lambda_1 u_1 + \mu_1 |u_1|^{p_1 -2}u_1 + \beta…
By applying a simple symmetry reduction on a two-layer liquid model, a nonlocal counterpart of it is obtained. Then a general form of nonlocal nonlinear Schrodinger (NNLS) equation with shifted parity, charge-conjugate and delayed time…
In this paper, we consider the longtime dynamics of the solutions to focusing energy-critical Schr\"odinger equation with a defocusing energy-subcritical perturbation term under a ground state energy threshold in four spatial dimension.…
In this paper we consider the nonlinear one-dimensional time-dependent Schroedinger equation with a periodic potential and a local perturbation. In the limit of large periodic potential the time behavior of the wavefunction can be…
There has been recent interest in the relaxational modes of small-scale fully connected systems of aligning self-propelled particles (Spera et al., Phys. Rev. Lett. {\bf 132}: 078301 (2024)). We revisit the classical connection between…
We study the nonlinear Schrodinger equations with a linear potential. A change of variables makes it possible to deduce results concerning finite time blow up and scattering theory from the case with no potential.
In this paper, we study the defocusing nonlinear Schr\"{o}dinger equation with a locally distributed damping on a smooth bounded domain as well as on the whole space and on an exterior domain. We first construct approximate solutions using…
We consider a class of nonlinear Schr\"odinger equation in two space dimensions with an attractive potential. The nonlinearity is local but rather general encompassing for the first time both subcritical and supercritical (in $L^2$)…
We demonstrate that stabilization of solitons of the multidimensional Schrodinger equation with a cubic nonlinearity may be achieved by a suitable periodic control of the nonlinear term. The effect of this control is to stabilize the…
By introducing and solving two correlative constrained variational problems as well as spectrum analysis, an approach to fix soliton frequency from the prescribed mass for nonlinear Schr\"odinger equations is found, and an open problem in…