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Search: MSC category 35Q55 ( NLS-like equations (nonlinear Schrodinger) [See also 37K10] )

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1. CJM 2011 (vol 63 pp. 1201)

Abou Salem, Walid K. ; Sulem, Catherine
 Resonant Tunneling of Fast Solitons through Large Potential Barriers We rigorously study the resonant tunneling of fast solitons through large potential barriers for the nonlinear SchrÃ¶dinger equation in one dimension. Our approach covers the case of general nonlinearities, both local and Hartree (nonlocal). Keywords:nonlinear Schroedinger equations, external potential, solitary waves, long time behavior, resonant tunnelingCategories:37K40, 35Q55, 35Q51

2. CJM 2008 (vol 60 pp. 1168)

Taylor, Michael
 Short Time Behavior of Solutions to Linear and Nonlinear Schr{Ã¶dinger Equations We examine the fine structure of the short time behavior of solutions to various linear and nonlinear Schr{\"o}dinger equations $u_t=i\Delta u+q(u)$ on $I\times\RR^n$, with initial data $u(0,x)=f(x)$. Particular attention is paid to cases where $f$ is piecewise smooth, with jump across an $(n-1)$-dimensional surface. We give detailed analyses of Gibbs-like phenomena and also focusing effects, including analogues of the Pinsky phenomenon. We give results for general $n$ in the linear case. We also have detailed analyses for a broad class of nonlinear equations when $n=1$ and $2$, with emphasis on the analysis of the first order correction to the solution of the corresponding linear equation. This work complements estimates on the error in this approximation. Categories:35Q55, 35Q40

3. CJM 2002 (vol 54 pp. 1065)

Hayashi, Nakao; Naumkin, Pavel I.
 Large Time Behavior for the Cubic Nonlinear SchrÃ¶dinger Equation We consider the Cauchy problem for the cubic nonlinear Schr\"odinger equation in one space dimension $$\begin{cases} iu_t + \frac12 u_{xx} + \bar{u}^3 = 0, & \text{t \in \mathbf{R}, x \in \mathbf{R},} \\ u(0,x) = u_0(x), & \text{x \in \mathbf{R}.} \end{cases} \label{A}$$ Cubic type nonlinearities in one space dimension heuristically appear to be critical for large time. We study the global existence and large time asymptotic behavior of solutions to the Cauchy problem (\ref{A}). We prove that if the initial data $u_0 \in \mathbf{H}^{1,0} \cap \mathbf{H}^{0,1}$ are small and such that $\sup_{|\xi|\leq 1} |\arg \mathcal{F} u_0 (\xi) - \frac{\pi n}{2}| < \frac{\pi}{8}$ for some $n \in \mathbf{Z}$, and $\inf_{|\xi|\leq 1} |\mathcal{F} u_0 (\xi)| >0$, then the solution has an additional logarithmic time-decay in the short range region $|x| \leq \sqrt{t}$. In the far region $|x| > \sqrt{t}$ the asymptotics have a quasi-linear character. Category:35Q55

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