## 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:

 [1] Mathematical Methods of Classical Mechanics, Springer-Verlag, 2nd ed., 1980. [2] Grillakis, J. of Func. Anal. 74 pp 160– (1987) [3] Grillakis, Comm. Pure Appl. Math. [4] Jones, J. Diff. Eq. [5] and , Methods of Modern Mathematical Physics, Vol. IV, Academic Press, 1978. [6] Topics in Differential and Integral Equations and Operator Theory, OT7 Birkhäuser, 1983, pp. 1–98. [7] and , Hill’s Equation, Dover, 1979. [8] Weinstein, Proc. of Symp. in Pure Math. Vol. 36 (1980) [9] Jones, SIAM J. Math. Anal. 17 (1986)
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