##
**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:
\[
i\partial_t u-\partial^2_xu+ V(x)u+ \varepsilon {\partial H\over\partial\overline u}=0,\tag{1}
\]

\[ u_{tt}- u_{xx}+ V(x)u+ \varepsilon f(u)=0\tag{2} \] 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 \[ |a_{j_1}\lambda_{j_1}+\cdots+ a_{j_r}\lambda_{j_r}|\geq\max (j^{-C(r)}_1,J^{-10r})\tag{3} \] for all \(j_1<\cdots< j_r<J\), \(a_{j_i}\in\mathbb{Z}\), \(a_{j_1}\neq 0\), \(\sum^r_{i=1}|a_{j_i}|\leq r\) 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 \[ |a_{j_1}\lambda_{j_1}+\cdots+ a_{j_r}\lambda_{j_r}|> J^{-30r}\tag{4} \] for \(j_1< j_2<\cdots< j_r<J\), \(J\) is large, \(a_j\in\mathbb{Z}\), \(0<\sum^r_{i=1}|a_{j_i}|< r\). For these typical potentials, i.e. potentials satisfying (3) and (4), the first main result states the following.

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.

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 \[ u_\varepsilon(x,t)= \sum^\infty_{j=1} \sum_{n\in\Pi_\infty\mathbb{Z}}\widehat u(j,n)\varphi_j(x) e^{i<n,\lambda'>t}.\tag{5} \] This solution is constructed by the meth;od developed before by the same author as a small \(\varepsilon\)-perturbation of the (unperturbed) solution \[ u_0(x,t)= \sum^\infty_{j=1} a_j\varphi_j(x)\cos\lambda_jt.\tag{6} \] 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 \(\Pi_\infty\mathbb{Z}\), \(\Pi_\infty \mathbb{Z}\) being the space of finite sequences of integers).

\[ u_{tt}- u_{xx}+ V(x)u+ \varepsilon f(u)=0\tag{2} \] 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 \[ |a_{j_1}\lambda_{j_1}+\cdots+ a_{j_r}\lambda_{j_r}|\geq\max (j^{-C(r)}_1,J^{-10r})\tag{3} \] for all \(j_1<\cdots< j_r<J\), \(a_{j_i}\in\mathbb{Z}\), \(a_{j_1}\neq 0\), \(\sum^r_{i=1}|a_{j_i}|\leq r\) 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 \[ |a_{j_1}\lambda_{j_1}+\cdots+ a_{j_r}\lambda_{j_r}|> J^{-30r}\tag{4} \] for \(j_1< j_2<\cdots< j_r<J\), \(J\) is large, \(a_j\in\mathbb{Z}\), \(0<\sum^r_{i=1}|a_{j_i}|< r\). For these typical potentials, i.e. potentials satisfying (3) and (4), the first main result states the following.

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.

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 \[ u_\varepsilon(x,t)= \sum^\infty_{j=1} \sum_{n\in\Pi_\infty\mathbb{Z}}\widehat u(j,n)\varphi_j(x) e^{i<n,\lambda'>t}.\tag{5} \] This solution is constructed by the meth;od developed before by the same author as a small \(\varepsilon\)-perturbation of the (unperturbed) solution \[ u_0(x,t)= \sum^\infty_{j=1} a_j\varphi_j(x)\cos\lambda_jt.\tag{6} \] 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 \(\Pi_\infty\mathbb{Z}\), \(\Pi_\infty \mathbb{Z}\) being the space of finite sequences of integers).

Reviewer: V.Georgiev (Sofia)

### MSC:

35B15 | Almost and pseudo-almost periodic solutions to PDEs |

35B30 | Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs |

35Q55 | NLS equations (nonlinear Schrödinger equations) |

35L70 | Second-order nonlinear hyperbolic equations |

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\textit{J. Bourgain}, Geom. Funct. Anal. 6, No. 2, 201--230 (1996; Zbl 0872.35007)

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