Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

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

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]