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Analysis of the linearization around a critical point of an infinite- dimensional Hamiltonian system. (English) Zbl 0731.35010

This paper deals with the linearized stability of stationary solutions of PDEs which can be written in the following general form: \[ (1)\quad du/dt=JE'(u). \] Here E is a functional defined on some Hilbert space (“energy”); and \(JJ^*=-1\). Examples of PDEs that can be put in this form include the nonlinear Klein-Gordon and Schrödinger equations: \[ (2)\quad u_{tt}+i\omega u_ t-\Delta u+f(x,| u|^ 2)u=0 \]
\[ (3)\quad iu_ t-\Delta u+f(x,| u|^ 2)u=0. \] The linearized stability of a stationary \((E'(u)=0)\) solution is determined by the spectrum of the operator \(A=JE''(u)\). In this work the author confines himself to the frequently encountered case when \(A=\left( \begin{matrix} 0\\ L_ 2\end{matrix} \begin{matrix} L_ 1\\ 0\end{matrix} \right)\) with \(L_ 1,L_ 2\) differential operators (not necessarily scalar). In this case the eigenvalue problem \(Aq=\zeta q\) can be reduced to
\[ (4)\quad (R-zS)p=0\text{ with } z=-\zeta^ 2, \] and R,S are suitable restrictions of \(L_ 1\), \(L_ 2^{-1}\). Let N(R) be the number of negative eigenvalues of R. The case N(R)\(\neq N(S)\) has already been analyzed in literature; so the author concentrates on the case \(N(R)=N(S)\). A method is developed for locating eigenvalues z of the problem (4) that are either unstable or satisfy \[ z>0;\quad S(z)=sign<Sp,p>\quad <0. \] (The latter property isolates stable eigenvalues which, however, become unstable when introducing a slight dissipation to the PDE (1).)
Next the structural stability is examined in the situation with an eigenvalue embedded in the continuous spectrum. (The operator A is said to be structurally unstable if there exists an arbitrarily small perturbation of A that has complex eigenvalues.) Finally, certain instability criteria are developed and applied to specific examples, namely to the radial-symmetric nodal solutions of eqs. (2), (3).

MSC:

35B35 Stability in context of PDEs
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35Q55 NLS equations (nonlinear Schrödinger equations)
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References:

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