In this paper we prove long time existence for a large class of fully nonlinear, reversible and parity preserving Schrödinger equations on the one dimensional torus. We show that, if some non-resonance conditions are fulfilled, for any N ∊ ℕ and for any initial condition, which is even in x and size ε in an appropriate Sobolev space, the lifespan of the solution is of order ε-N. After a paralinearization of the equation we perform several para-differential changes of variables which diagonalize the system up to a very regularizing term. Once achieved the diagonalization, we construct modified energies for the solution by means of Birkhoff normal forms techniques.

Feola, R., & Iandoli, F. (2021). Long time existence for fully nonlinear NLS with small Cauchy data on the circle. ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA. CLASSE DI SCIENZE, 22(1), 109-182 [10.2422/2036-2145.201811_003].

Long time existence for fully nonlinear NLS with small Cauchy data on the circle

Feola R.;
2021

Abstract

In this paper we prove long time existence for a large class of fully nonlinear, reversible and parity preserving Schrödinger equations on the one dimensional torus. We show that, if some non-resonance conditions are fulfilled, for any N ∊ ℕ and for any initial condition, which is even in x and size ε in an appropriate Sobolev space, the lifespan of the solution is of order ε-N. After a paralinearization of the equation we perform several para-differential changes of variables which diagonalize the system up to a very regularizing term. Once achieved the diagonalization, we construct modified energies for the solution by means of Birkhoff normal forms techniques.
Feola, R., & Iandoli, F. (2021). Long time existence for fully nonlinear NLS with small Cauchy data on the circle. ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA. CLASSE DI SCIENZE, 22(1), 109-182 [10.2422/2036-2145.201811_003].
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11590/396993
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