We consider the nonlinear wave equation Utt - C2Uxx = εψ(u), u(t,0) = u(t,π) = 0, (*) where ψ is an analytic function satisfying ψ(0) = 0 and ψ'(0) ≠ 0. For each value of the harmonic energy E (i.e. energy when ε = 0), we consider the closed curve which is the phase space trajectory of the linear mode with lowest frequency. If the perturbation parameter ε and the harmonic energy E are small enough, then, near γE we construct, by means of a suitable "simplified equation", a closed curve γ= γE,ε with the following property: provided the distance between the initial datum and such a curve is small enough, the corresponding solution remains close to γE,ε up to times exponentially long with 1/ε; one cannot expect the curves γE,ε to be trajectories of solutions of (*). Such curves depend smoothly on the parameters E and ε. The above result is deduced from a general theorem on generic Hamiltonian perturbations of completely resonant linear systems.

A property of exponential stability in nonlinear wave equations near the fundamental linear mode / D. Bambusi, N.N. Nekhoroshev. - In: PHYSICA D-NONLINEAR PHENOMENA. - ISSN 0167-2789. - 122:1-4(1998), pp. 73-104. [10.1016/S0167-2789(98)00169-9]

A property of exponential stability in nonlinear wave equations near the fundamental linear mode

D. Bambusi
Primo
;
1998

Abstract

We consider the nonlinear wave equation Utt - C2Uxx = εψ(u), u(t,0) = u(t,π) = 0, (*) where ψ is an analytic function satisfying ψ(0) = 0 and ψ'(0) ≠ 0. For each value of the harmonic energy E (i.e. energy when ε = 0), we consider the closed curve which is the phase space trajectory of the linear mode with lowest frequency. If the perturbation parameter ε and the harmonic energy E are small enough, then, near γE we construct, by means of a suitable "simplified equation", a closed curve γ= γE,ε with the following property: provided the distance between the initial datum and such a curve is small enough, the corresponding solution remains close to γE,ε up to times exponentially long with 1/ε; one cannot expect the curves γE,ε to be trajectories of solutions of (*). Such curves depend smoothly on the parameters E and ε. The above result is deduced from a general theorem on generic Hamiltonian perturbations of completely resonant linear systems.
Settore MAT/07 - Fisica Matematica
1998
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/67878
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