Aspects of long time behaviour of solutions of nonlinear Hamiltonian evolution equations
This paper concentrates on the long time asymptotic behaviour for Hamiltonian nonlinear Schrödinger equations of the form: \[ iu_{t}+\Delta u + V(x)u+\frac{\partial }{\partial\overline{u}} G(u,\overline{u} , t,x) = 0, x\in {\mathbf T}^d, \] where \(V(x)\) is a real valued potential, and \(G\) is a polynomial in \(u,\overline{u} \). Global well-posedness has been addressed in previous papers, without further insight into the behaviour of the solutions. Here, given the dimension \(d\) and the degree of the polynomial \(G\) , the author shows that bounded solutions of (NLSE) are not uniformly asymptotically stable in \(L^{2}\) for \(t\rightarrow\infty \). The same result holds in \(H^{s}\) for \(s>1/2\), without any restriction on the degree of the perturbation \(G\). These results are based on a non-squeezing property of the flow \(S_{t}\), governed by (NLSE), which was first discovered by \textit{S. B. Kuksin} [Commun. Math. Phys. 167, No. 7, 531-552 (1995; Zbl 0827.35121)] for infinite dimensional Hamiltonian systems in the special case \(S_{t}\) = linear operator + compact smooth operator. The present article shows the same type of behaviour, but for the interesting cases, such as the cubic NLSE, where the map \(S_{t}\) does not fit in the previous description. Even more, the author gives an example of Hamiltonian NLSE with solutions having arbitrarily large \(H^{s}\) norms. A concrete example is a Hamiltonian perturbation of the linear Schrödinger equation, such that \(\limsup |S_{t} u|{H^{s_0}} = \infty \) for some fixed exponent \(s_{0}\). The construction is related to an infinite dimensional version of the Lyapunov's theorem in KAM theory. The methods used are related to some restriction theorems for the Fourier transform and Strichartz type inequalities.
scientific article; zbMATH DE number 815715