Self-similar solutions for the Schrödinger map equation

We study in this article the equivariant Schrödinger map equation in dimension 2, from the Euclidean plane to the sphere. A family of self-similar solutions is constructed; this provides an example of regularity breakdown for the Schrödinger map. These solutions do not have finite energy, and hence do not fit into the usual framework for solutions. For data of infinite energy but small in some norm, we build up associated global solutions.     

Expansion with respect to ħ for bound states of the Schrödinger equation

A WKB-complementing expansion for bound states of the radial Schrödinger equation is discussed. A recursive method for calculating the quantum corrections of any order to the energy of the classical motion is presented. The use of quantization conditions makes it possible to write down recursion relations in an equally simple form for the ground and radially excited states. The connection between the approach and the 1/N expansion is considered. It is shown that the method can also be used for analysis in thel, E) plane in the form of a expansion for Regge trajectories.Dnepropetrovsk State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 90, No. 2, pp. 208–217, February, 1992.     

Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation

We consider the critical nonlinear Schrödinger equation $iu_{t} = -\Delta u-|u|^{4/N}$ with initial condition u(0, x) = u₀.For u₀$\in$H¹, local existence in time of solutions on an interval [0, T) is known, and there exist finite time blow-up solutions, that is u₀ such that $\textrm{lim} _{t\uparrow T <+\infty}|\nabla u(t)|_{L^{2}}=+\infty$. This is the smallest power in the nonlinearity for which blow-up occurs, and is critical in this sense.The question we address is to control the blow-up rate from above for small (in a certain sense) blow-up solutions with negative energy. In a previous paper [MeR], we established some blow-up properties of (NLS) in the energy space which implied a control $|\nabla u(t)|_{L^{2}} \leq C \frac{|\ln(T-t)|^{N/4}}{\sqrt{T-t}}$ and removed the rate of the known explicit blow-up solutions which is $\frac{C}{T-t}$.In this paper, we prove the sharp upper bound expected from numerics as$|\nabla u(t)|_{L^{2}} \leq C \left(\frac{\ln|\ln(T-t)|}{T-t} \right)^{1/2}$by exhibiting the exact geometrical structure of dispersion for the problem.     

Positive solutions of a Schrödinger equation with critical nonlinearity

We study the nonlinear Schröodinger equation with critical exponent 2^*= 2 N/( N-2), N 4, where a 0, has a potential well. Using variational methods we establish existence and multiplicity of positive solutions which localize near the potential well for small and large.     

Modified scattering for the critical nonlinear Schrödinger equation

We consider the nonlinear Schrödinger equation

$i{u}_t+\mathrm{\Delta }u=\lambda |u{|}^{\frac{2}{N}}u$

in all dimensions $N\ge 1$, where $\lambda \in \mathbb{C}$ and $?\lambda \le 0$. We construct a class of initial values for which the corresponding solution is global and decays as $t\to \infty$, like $t^{?\frac{N}{2}}$ if $?\lambda =0$ and like ${(t\log ?t)}^{?\frac{N}{2}}$ if $?\lambda <0$. Moreover, we give an asymptotic expansion of those solutions as $t\to \infty$. We construct solutions that do not vanish, so as to avoid any issue related to the lack of regularity of the nonlinearity at $u=0$. To study the asymptotic behavior, we apply the pseudo-conformal transformation and estimate the solutions by allowing a certain growth of the Sobolev norms which depends on the order of regularity through a cascade of exponents.     

The structure of solutions of the Schrödinger equation

The goal of the paper is to study the structure of the eigenfunctions of the one-dimensional Schrödinger equation from the point of view of the Euler theorem. It turns out that analog of exponent is exponentially increasing solution. Sometimes linear combinations of such solutions cancel each other at infinity and then we obtain an eigenfunction from L₂(R¹).     

Radial functions and regularity of solutions to the Schrödinger equation

Letf be a radial function and setT ^* f(x)=sup_0<t<1 |T _t f(x)|, x ∈ ?ⁿ, n≥2, where(T_t f)^ (ξ)=e ^it|ξ|a \(\hat f\) (ξ),a > 1. We show that, ifB is the ball centered at the origin, of radius 100, then \(\int\limits_B {|T^ * f(x)|} dx \leqslant c(\int {|\hat f(\xi )|^2 (l + |\xi |^s )ds} )^{1/2} \) if and only ifs≥1/4.     

The elliptic-in-t solutions of the nonlinear Schrödinger equation

Four various anzatzes of the Krichever curves for the elliptic-in-t solutions of the nonlinear Schrödinger equation are considered. An example is given.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 107, No. 2, pp. 188–200, May, 1996.Translated by V. I. Serdobol'skii.     

Asymptotic behaviour of solutions of the difference Schrödinger equation

We use the method of averaging and the discrete analogue of Levinson's theorem to construct the asymptotics for solutions of the difference Schrödinger equation. Moreover, we present the general form for the averaging change of variable.     

Stability of solutions for nonlinear Schrdinger equations in critical spaces

We consider the Cauchy problem for nonlinear Schrdinger equation iut + Δu = ±|u|pu,4/d< p <4 /d-2 in high dimensions d 6. We prove the stability of solutions in the critical space H˙xsp , where sp = d/2-p/2 .     

Exponentially small asymptotics of solutions to the defocusing nonlinear Schrödinger equation

The matrix Riemann-Hilbert factorization approach is used to derive the leading-order, exponentially small asymptotics as t → ± ∞ such that x/t ∼ O(1) of solutions to the Cauchy problem for the defocusing nonlinear Schrödinger equation, i∂_tu + ∂_x²u − 2(|u|² − 1)u = 0, with finite density initial data u(x,0) = _x→±∞exp(i(1 ∓ 1)φ/2)(1+o(1)), φ ϵ [0, 2π).     

Global existence of small solutions for the fourth-order nonlinear Schrödinger equation

We consider the Cauchy problem for the fourth-order nonlinear Schrödinger equation

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where \(n=1,2\). We prove global existence of small solutions under the growth condition of \(f\left( u\right) \) satisfying \(\left| \partial _{u}^{j}f\left( u\right) \right| \le C\left| u\right| ^{p-j},\) where \(p>1+\frac{4}{n},0\le j\le 3\).

     

Modulation of localized solutions for the Schrödinger equation with logarithm nonlinearity

We investigate the presence of localized analytical solutions of the Schrödinger equation with logarithm nonlinearity. After including inhomogeneities in the linear and nonlinear coefficients, we use similarity transformation to convert the nonautonomous nonlinear equation into an autonomous one, which we solve analytically. In particular, we study stability of the analytical solutions numerically.