# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Construction of approximative and almost periodic solutions of perturbed linear Schrödinger and wave equations. (English) Zbl 0872.35007

The work is devoted to the study of the following two perturbations of the Schrödinger and wave equations, respectively:

$i{\partial }_{t}u-{\partial }_{x}^{2}u+V\left(x\right)u+\epsilon \frac{\partial H}{\partial \overline{u}}=0,\phantom{\rule{2.em}{0ex}}\left(1\right)$
${u}_{tt}-{u}_{xx}+V\left(x\right)u+\epsilon f\left(u\right)=0\phantom{\rule{2.em}{0ex}}\left(2\right)$

under Dirichlet boundary conditions. Here $H\left(u,\overline{u}\right)$ and $f\left(u\right)$ are assumed to be polynomials, while $V\left(x\right)$ is a periodic potential. The main goal of the work is to find almost periodic solutions of these partial differential equations and evaluate their asymptotic behaviour as $\epsilon \to 0$. This fact is established for suitable potentials satisfying nonresonance properties. More precisely, for generic $V\left(x\right)$ it is shown that

$|{a}_{{j}_{1}}{\lambda }_{{j}_{1}}+\cdots +{a}_{{j}_{r}}{\lambda }_{{j}_{r}}|\ge max\left({j}_{1}^{-C\left(r\right)},{J}^{-10r}\right)\phantom{\rule{2.em}{0ex}}\left(3\right)$

for all ${j}_{1}<\cdots <{j}_{r}, ${a}_{{j}_{i}}\in ℤ$, ${a}_{{j}_{1}}\ne 0$, ${\sum }_{i=1}^{r}|{a}_{{j}_{i}}|\le r$ and ${j}_{1}$ sufficiently large. Here $\left\{{\lambda }_{j}\right\}$ is the Dirichlet spectrum for the operator $-{\partial }_{x}^{2}+V\left(x\right)$. Moreover, for generic $V$ one has

$|{a}_{{j}_{1}}{\lambda }_{{j}_{1}}+\cdots +{a}_{{j}_{r}}{\lambda }_{{j}_{r}}|>{J}^{-30r}\phantom{\rule{2.em}{0ex}}\left(4\right)$

for ${j}_{1}<{j}_{2}<\cdots <{j}_{r}, $J$ is large, ${a}_{j}\in ℤ$, $0<{\sum }_{i=1}^{r}|{a}_{{j}_{i}}|. For these typical potentials, i.e. potentials satisfying (3) and (4), the first main result states the following.

Theorem. Suppose that $V\left(x\right)$ is an even real periodic potential and $H$ is a polynomial of the form $H\left(|u{|}^{2}\right)$. Let $u\left(0\right)$ be a smooth initial function for $t=0$. Then the solution $u$ of (1) will be, for times $|t|<{\epsilon }^{-M}$, an ${\epsilon }^{1/2}$-perturbation of the unperturbed solution with appropriate frequency adjustment. Here $M>0$ may be taken to be any fixed number.

For the case of the wave equation (2) the nonlinear term $f\left(u\right)$ is assumed to be an odd polynomial function of $u$, $f\left(u\right)=O\left(|u{|}^{3}\right)$. Denote by $\left\{{\mu }_{j}\right\}$ and $\left\{{\varphi }_{j}\right\}$ the Dirichlet spectrum and the eigenfunctions of $-{\partial }_{x}^{2}+V\left(x\right)$. Setting ${\mu }_{i}={\lambda }_{j}^{2}$, the author looks for a solution of (2) in the form

${u}_{\epsilon }\left(x,t\right)=\sum _{j=1}^{\infty }\sum _{n\in {{\Pi }}_{\infty }ℤ}\stackrel{^}{u}\left(j,n\right){\varphi }_{j}\left(x\right){e}^{it}·\phantom{\rule{2.em}{0ex}}\left(5\right)$

This solution is constructed by the meth;od developed before by the same author as a small $\epsilon$-perturbation of the (unperturbed) solution

${u}_{0}\left(x,t\right)=\sum _{j=1}^{\infty }{a}_{j}{\varphi }_{j}\left(x\right)cos{\lambda }_{j}t·\phantom{\rule{2.em}{0ex}}\left(6\right)$

Under the natural assumption that $\left\{{a}_{i}\right\}$ tends to 0 sufficiently rapidly, for typical real analytic potentials $V$, the existence of an almost periodic solution of (2) is established. In addition this solution satisfies the properties $\stackrel{^}{u}\left(j,n\right)=\stackrel{^}{u}\left(j,-n\right)$, ${\lambda }_{j}^{\text{'}}={\lambda }_{j}+O\left(\epsilon /j\right)$ (uniformly in $j$) is the perturbed frequency, $\stackrel{^}{u}\left(j,{e}_{j}\right)=\stackrel{^}{u}\left(j,-{e}_{j}\right)=\frac{1}{2}{a}_{j}$ (${e}_{j}=j$-unit vector in ${{\Pi }}_{\infty }ℤ$, ${{\Pi }}_{\infty }ℤ$ being the space of finite sequences of integers).

##### MSC:
 35B15 Almost and pseudo-almost periodic solutions of PDE 35B30 Dependence of solutions of PDE on initial and boundary data, parameters 35Q55 NLS-like (nonlinear Schrödinger) equations 35L70 Nonlinear second-order hyperbolic equations
##### Keywords:
periodic potential; nonresonance properties
##### References:
 [1] G. Benettin, J. Fröhlich, A. Giorgilli, A Nekhoroshev-type theorem for Hamiltonian systems with infinitely many degrees of freedom, CMP 119 (1989), 95–108. [2] J. Bourgain, Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear PDE, International Math. Research Notices 11 (1994), 475–497. · Zbl 0817.35102 · doi:10.1155/S1073792894000516 [3] L. Chierchia, P. Perfetti, Maximal Almost-periodic Solutions for Langiangian Equations on Infinite Dimensional Tori, in "Seminar on Dynamical Systems", Progress in Nonlinear Differential equations and Their Applications 12, Birkhäuser, 1994. [4] W. Craig, E. Wayne, Newton’s method and periodic solutions of nonlinear wave equations, Comm. on Pure and Applied Math 46:11 (1993), 1405–1498. · Zbl 0794.35104 · doi:10.1002/cpa.3160461102 [5] J. Fröhlich, T. Spencer, E. Wayne, Localization in disordered, nonlinear dynamical systems, J. Stat. Ph. 42 (1986), 257–275. [6] N.N. Nekhoroshev, An exponential estimate of the time of stability of nearly integrable Hamiltonian systems, Russ. Math. Surveys 32 (1977), 1–65. · Zbl 0389.70028 · doi:10.1070/RM1977v032n06ABEH003859 [7] J. Pöschel, E. Trubowitz, Inverse Spectral Theory Boston, Academic Press, 1987. [8] S. Kuksin, J. Pöschel, Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödiger equation, Annals of Math. 143:1 (1996), 149–179. · Zbl 0847.35130 · doi:10.2307/2118656