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Zbl 0872.35007
Bourgain, Jean
Construction of approximative and almost periodic solutions of perturbed linear Schrödinger and wave equations.
(English)
[J] Geom. Funct. Anal. 6, No.2, 201-230 (1996). ISSN 1016-443X; ISSN 1420-8970/e

The work is devoted to the study of the following two perturbations of the Schrödinger and wave equations, respectively: $$i\partial_t u-\partial^2_xu+ V(x)u+ \varepsilon {\partial H\over\partial\overline u}=0,\tag1$$ $$u_{tt}- u_{xx}+ V(x)u+ \varepsilon f(u)=0\tag2$$ under Dirichlet boundary conditions. Here $H(u,\overline u)$ and $f(u)$ are assumed to be polynomials, while $V(x)$ 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 $\varepsilon\to 0$. This fact is established for suitable potentials satisfying nonresonance properties. More precisely, for generic $V(x)$ it is shown that $$|a_{j_1}\lambda_{j_1}+\cdots+ a_{j_r}\lambda_{j_r}|\ge\max (j^{-C(r)}_1,J^{-10r})\tag 3$$ for all $j_1<\cdots< j_r<J$, $a_{j_i}\in\bbfZ$, $a_{j_1}\ne 0$, $\sum^r_{i=1}|a_{j_i}|\le r$ and $j_1$ sufficiently large. Here $\{\lambda_j\}$ is the Dirichlet spectrum for the operator $-\partial^2_x+ V(x)$. Moreover, for generic $V$ one has $$|a_{j_1}\lambda_{j_1}+\cdots+ a_{j_r}\lambda_{j_r}|> J^{-30r}\tag4$$ for $j_1< j_2<\cdots< j_r<J$, $J$ is large, $a_j\in\bbfZ$, $0<\sum^r_{i=1}|a_{j_i}|< r$. For these typical potentials, i.e. potentials satisfying (3) and (4), the first main result states the following.\par Theorem. Suppose that $V(x)$ is an even real periodic potential and $H$ is a polynomial of the form $H(|u|^2)$. Let $u(0)$ be a smooth initial function for $t=0$. Then the solution $u$ of (1) will be, for times $|t|<\varepsilon^{-M}$, an $\varepsilon^{1/2}$-perturbation of the unperturbed solution with appropriate frequency adjustment. Here $M>0$ may be taken to be any fixed number.\par For the case of the wave equation (2) the nonlinear term $f(u)$ is assumed to be an odd polynomial function of $u$, $f(u)= O(|u|^3)$. Denote by $\{\mu_j\}$ and $\{\varphi_j\}$ the Dirichlet spectrum and the eigenfunctions of $-\partial^2_x+ V(x)$. Setting $\mu_i=\lambda^2_j$, the author looks for a solution of (2) in the form $$u_\varepsilon(x,t)= \sum^\infty_{j=1} \sum_{n\in\Pi_\infty\bbfZ}\widehat u(j,n)\varphi_j(x) e^{i<n,\lambda'>t}.\tag5$$ This solution is constructed by the meth;od developed before by the same author as a small $\varepsilon$-perturbation of the (unperturbed) solution $$u_0(x,t)= \sum^\infty_{j=1} a_j\varphi_j(x)\cos\lambda_jt.\tag6$$ Under the natural assumption that $\{a_i\}$ 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 $\widehat u(j,n)= \widehat u(j,-n)$, $\lambda_j'=\lambda_j+O (\varepsilon/j)$ (uniformly in $j$) is the perturbed frequency, $\widehat u(j,e_j)=\widehat u(j,-e_j)={1\over 2}a_j$ ($e_j=j$-unit vector in $\Pi_\infty\bbfZ$, $\Pi_\infty \bbfZ$ being the space of finite sequences of integers).
[V.Georgiev (Sofia)]
MSC 2000:
*35B15 Almost periodic solutions of PDE
35B30 Dependence of solutions of PDE on initial and boundary data
35Q55 NLS-like (nonlinear Schroedinger) equations
35L70 Second order nonlinear hyperbolic equations

Keywords: periodic potential; nonresonance properties

Cited in: Zbl 1185.81073 Zbl 0967.35014

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