Blow-up of Solutions of the Cauchy Problem for a Nonlinear Schrödinger Evolution Equation

The solution of the Cauchy problem for a nonlinear Schrödinger evolution equation with certain initial data is proved to blow up in a finite time, which is estimated from above. Additionally, lower bounds for the blow-up rate are obtained in some norms.     

On the Collapse of Solutions of the Cauchy Problem for the Cubic Schrödinger Evolution Equation

It is proved that, for some initial data, the solutions of the Cauchy problem for the cubic Schrödinger evolution equation blow up in finite time whose exact value is estimated from above. In addition, lower bounds for the blow-up rate of the solution in certain norms are obtained.

     

On Multiwave Solutions of One Nonlinear Schrödinger Equation

Differential Equations - We consider a nonlinear Schrödinger equation arising in a number of physical problems. It is shown that when the real part is separated in this equation, there arises...     

Solutions of the Discrete Nonlinear Schrödinger Equation with a Trap

Theoretical and Mathematical Physics - We obtain solutions of the discrete nonlinear Schrödinger equation with an impurity center in two ways. In the first of them, we construct the wave...     

Finite Morse Index Solutions of a Nonlinear Schrödinger Equation

Acta Mathematica Sinica, English Series - We prove Liouville type theorems for stable and finite Morse index H loc 1 ∩ L loc ∞ solutions of the nonlinear Schrödinger equation...     

Symmetries of the Discrete Nonlinear Schrödinger Equation

The Lie algebra L(h) of point symmetries of a discrete analogue of the nonlinear Schrödinger equation (NLS) is described. In the continuous limit, the discrete equation is transformed into the NLS, while the structure of the Lie algebra changes: a contraction occurs with the lattice spacing h as the contraction parameter. A five-dimensional subspace of L(h), generated by both point and generalized symmetries, transforms into the five-dimensional point symmetry algebra of the NLS.     

Sobolev Stability of Plane Wave Solutions to the Nonlinear Schrödinger Equation

This article explores the questions of long time orbital stability in high order Sobolev norms of plane wave solutions to the NLSE in the defocusing case.     

Multiple Solutions for a Nonlinear Schrödinger Equation with Magnetic Fields

We study a nonlinear Schrödinger equation in presence of a magnetic field and relate the number of solutions with the topology of the set where the potential attains its minimum value. In the proofs we apply variational methods, penalization techniques and Ljusternik–Schnirelmann theory.     

Breathers for the Discrete Nonlinear Schrödinger Equation with Nonlinear Hopping

We discuss the existence of breathers and lower bounds on their power, in nonlinear Schrödinger lattices with nonlinear hopping. Our methods extend from a simple variational approach to fixed-point arguments, deriving lower bounds for the power which can serve as a threshold for the existence of breather solutions. Qualitatively, the theoretical results justify non-existence of breathers below the prescribed lower bounds of the power which depend on the dimension, the parameters of the lattice as well as of the frequency of breathers. In the case of supercritical power nonlinearities we investigate the interplay of these estimates with the optimal constant of the discrete interpolation inequality. Improvements of the general estimates, taking into account the localization of the true breather solutions are derived. Numerical studies in the one-dimensional lattice corroborate the theoretical bounds and illustrate that in certain parameter regimes of physical significance, the estimates can serve as accurate predictors of the breather power and its dependence on the various system parameters.     

Dispersive Estimates of Solutions to the Schrödinger Equation

We prove time decay L¹ → L^∞ estimates for the Schr?dinger group e^{it(−Δ + V)} for real-valued potentials satisfying V (x) = O (|x|^−δ), |x| ≫ 1, with δ > 5/2. Communicated by Bernard Helffer submitted 27/11/04, accepted 29/04/05     

Final State Problem for the Cubic Nonlinear Schrödinger Equation with Repulsive Delta Potential

We consider the asymptotic behavior in time of solutions to the cubic nonlinear Schrödinger equation with repulsive delta potential (δ-NLS). We shall prove that for a given small asymptotic profile u _ap , there exists a solution u to (δ-NLS) which converges to u _ap in L ²(?) as t → ∞. To show this result we exploit the distorted Fourier transform associated to the Schrödinger equation with delta potential.     

Absence of Global Solutions of a Mixed Problem for a Schrödinger-Type Nonlinear Evolution Equation

We study the problem of the absence of global solutions of the first mixed problem for one nonlinear evolution equation of Schrödinger type. We prove that global solutions of the studied problem are absent for “sufficiently large” values of the initial data.     

Large-Order Asymptotics for Multiple-Pole Solitons of the Focusing Nonlinear Schrödinger Equation

Journal of Nonlinear Science - We analyze the large-n behavior of soliton solutions of the integrable focusing nonlinear Schrödinger equation with associated spectral data consisting of a...     

A Two-Grid Finite-Element Method for the Nonlinear Schrödinger Equation

In this paper, some two-grid finite element schemes are constructed for solving the nonlinear Schrödinger equation. With these schemes, the solution of the original problem is reduced to the solution of the same problem on a much coarser grid together with the solutions of two linear problems on a fine grid. We have shown, both theoretically and numerically, that our schemes are efficient and achieve asymptotically optimal accuracy.     

Formulas that Represent Cauchy Problem Solution for Momentum and Position Schrödinger Equation

In the paper we derive two formulas representing solutions of Cauchy problem for two Schrödinger equations: one-dimensional momentum space equation with polynomial potential, and multidimensional position space equation with locally square integrable potential. The first equation is a constant coefficients particular case of an evolution equation with derivatives of arbitrary high order and variable coefficients that do not change over time, this general equation is solved in the paper. We construct a family of translation operators in the space of square integrable functions and then use methods of functional analysis based on Chernoff product formula to prove that this family approximates the solution-giving semigroup. This leads us to some formulas that express the solution for Cauchy problem in terms of initial condition and coefficients of the equations studied.