Modified scattering for the critical nonlinear Schr\"odinger equation

We consider the nonlinear Schr\"odinger equation $iu_t + \Delta u= \lambda |u|^{\frac {2} {N}} u$ in all dimensions $N\ge 1$, where $\lambda \in {\mathbb C}$ and $\Im \lambda \le 0$. 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 $\Im \lambda =0$ and like $(t \log t)^{- \frac {N} {2}}$ if $\Im \lambda <0$. 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

Continuous dependence for NLS in fractional order spaces

We consider the Cauchy problem for the nonlinear Schr\"odinger equation $iu_t+ \Delta u+ \lambda |u|^\alpha u=0$ in $\R^N$, in the $H^s$-subcritical and critical cases $0<\alpha \le 4/(N-2s)$, where $0<s<N/2$. Local existence of solutions in $H^s$ is well known. However, even though the solution is constructed by a fixed-point technique, continuous dependence in $H^s$ does not follow from the contraction mapping argument. In this paper, assuming furthermore $s<1$, we show that the solution depends continuously on the initial value in the sense that the local flow is continuous $H^s \to H^s$. If, in addition, $\alpha \ge 1$ then the flow is Lipschitz. This completes previously known results concerning the cases $s=0,1,2$.Comment: Corrected typos. Simplified section 4. Results unchange

On the propagation of confined waves along the geodesics

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Standing waves of the complex Ginzburg-Landau equation

We prove the existence of nontrivial standing wave solutions of the complex Ginzburg-Landau equation $\phi_t = e^{i\theta} \Delta \phi + e^{i\gamma} |\phi |^\alpha \phi$ with periodic boundary conditions. Our result includes all values of $\theta$ and $\gamma$ for which $\cos \theta \cos \gamma >0$, but requires that $\alpha >0$ be sufficiently small

Finite-time blowup for a complex Ginzburg-Landau equation with linear driving

In this paper, we consider the complex Ginzburg--Landau equation $u_t = e^{i\theta} [\Delta u + |u|^\alpha u] + \gamma u$ on ${\mathbb R}^N$, where $\alpha >0$, $\gamma \in \R$ and $-\pi /2<\theta <\pi /2$. By convexity arguments we prove that, under certain conditions on $\alpha ,\theta ,\gamma$, a class of solutions with negative initial energy blows up in finite time