A Nekhoroshev type theorem for the derivative nonlinear Schrödinger equation. (English) Zbl 1437.37099
Summary: It is proved a Nekhoroshev type theorem for the derivative nonlinear Schrödinger equation in a Gevrey space. More precisely, we prove that if the norm of initial datum is equal to \(\varepsilon / 2\), then if \(\epsilon\) is small enough, the norm of the solution of the nonlinear Schrödinger equation above is bounded by \(2 \epsilon\) over a very long time interval of order \(e^{| \ln \varepsilon |^{1 + \beta}}\), where \(0 < \beta < \frac{ 1}{ 2}\) is arbitrary.
MSC:
| 37K55 | Perturbations, KAM theory for infinite-dimensional Hamiltonian and Lagrangian systems |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
Keywords:
unbounded perturbation; Nekhoroshev estimate; Hamiltonian system; derivative nonlinear Schrödinger equationReferences:
| [1] | Bambusi, D., Nekhoroshev theorem for small amplitude solutions in nonlinear Schrödinger equations, Math. Z., 230, 345-387 (1999) · Zbl 0928.35160 |
| [2] | Bambusi, D.; Grébert, B., Birkhoff normal form for partial differential equations with tame modulus, Duke Math. J., 135, 507-567 (2006) · Zbl 1110.37057 |
| [3] | Bambusi, D.; Delort, J. M.; Grébert, B.; Szeftel, J., Almost global existence for Hamiltonian semilinear Klein-Gordon equations with small Cauchy data on Zoll manifolds, Commun. Pure Appl. Math., 60, 11, 1665-1690 (2007) · Zbl 1170.35481 |
| [4] | Baldi, P.; Berti, M.; Haus, E.; Montalto, R., Time quasi-periodic gravity water waves in finite depth, Invent. Math., 214, 2, 739-911 (2018) · Zbl 1445.76017 |
| [5] | Benettin, G.; Fröhlich, J.; Giorgilli, A., A Nekhoroshev-type theorem for Hamiltonian systems with infinite many degrees of freedom, Commun. Math. Phys., 119, 1, 95-108 (1988) · Zbl 0825.58011 |
| [6] | Feola, R.; Iandoli, F., Local well-posedness for quasi-linear NLS with large Cauchy data on the circle, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 36, 1, 119-164 (2019) · Zbl 1430.35207 |
| [7] | Bernier, J.; Faou, E.; Grébert, B., Rational normal forms and stability of small solutions to nonlinear Schrödinger equations (2018), arXiv preprint |
| [8] | Berti, M.; Delort, J. M., Almost global existence of solutions for capillarity-gravity water waves equations with periodic spatial boundary conditions (2017), arXiv preprint |
| [9] | Biasco, L.; Massetti, J.; Procesi, M., Exponential stability estimates for the 1D NLS (2018) |
| [10] | Bourgain, J., Construction of approximative and almost periodic solutions of perturbed linear Schrödinger and wave equations, Geom. Funct. Anal., 6, 201-230 (1996) · Zbl 0872.35007 |
| [11] | Bourgain, J., Remark on stability and diffusion in high-dimensional Hamiltonian systems and partial differential equations, Ergod. Theory Dyn. Syst., 24, 1331-1357 (2004) · Zbl 1087.37056 |
| [12] | Bourgain, J., On invariant tori of full dimension for 1D periodic NLS, J. Funct. Anal., 229, 1, 62-94 (2005) · Zbl 1088.35061 |
| [13] | Delort, J. M., Long-time Sobolev stability for small solutions of quasi-linear Klein-Gordon equations on the circle, Trans. Am. Math. Soc., 361, 8, 4299-4365 (2009) · Zbl 1179.35195 |
| [14] | Delort, J. M.; Szeftel, J., Long time existence for small data nonlinear Klein-Gordon equations on tori and spheres, Int. Math. Res. Not., 37, 1897-1966 (2004) · Zbl 1079.35070 |
| [15] | Fang, D.; Han, Z.; Zhang, Q., Almost global existence for the semi-linear Klein-Gordon equation on the circle, J. Differ. Equ., 262, 9, 4610-4634 (2017) · Zbl 1361.35113 |
| [16] | Fang, D.; Zhang, Q., Long-time existence for semi-linear Klein-Gordon equations on tori, J. Differ. Equ., 249, 1, 151-179 (2010) · Zbl 1200.35189 |
| [17] | Faou, E.; Grébert, B., A Nekhoroshev-type theorem for the nonlinear Schrödinger equation on the torus, Anal. PDE, 6, 1243-1262 (2013) · Zbl 1284.35067 |
| [18] | Feola, R.; Giuliani, F.; Procesi, M., Reducible KAM tori for Degasperis-Procesi equation (2018) · Zbl 1446.35171 |
| [19] | Feola, R.; Giuliani, F.; Pasquali, S., On the integrability of Degasperis-Procesi equation: control of the Sobolev norms and Birkhoff resonances, J. Differ. Equ., 266, 6, 3390-3437 (2019) · Zbl 1407.35060 |
| [20] | Grébert, B.; Imekraz, R.; Paturel, E., Normal forms for semilinear quantum harmonic oscillators, Commun. Math. Phys., 291, 3, 763-798 (2009) · Zbl 1185.81073 |
| [21] | Pasquali, S., A Nekhoroshev type theorem for the nonlinear Klein-Gordon equation with potential, Discrete Contin. Dyn. Syst., Ser. B, 23, 9, 3573-3594 (2018) · Zbl 1410.35154 |
| [22] | Yuan, X. P.; Zhang, J., Long time stability of Hamiltonian partial differential equations, SIAM J. Math. Anal., 46, 5, 3176-3222 (2014) · Zbl 1405.37085 |
| [23] | Yuan, X. P.; Zhang, J., Averaging principle for the KdV equation with a small initial value, Nonlinearity, 29, 603-656 (2016) · Zbl 1417.37258 |
| [24] | Zhang, Q., Lifespan estimates for the semi-linear Klein-Gordon equation with a quadratic potential in dimension one, J. Differ. Equ., 261, 12, 6982-6999 (2016) · Zbl 1356.35146 |
| [25] | Zhang, Q., Long-time existence for semi-linear Klein-Gordon equations with quadratic potential, Commun. Partial Differ. Equ., 35, 4, 630-668 (2010) · Zbl 1201.35145 |
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