Comptes Rendus
no. 7-8
Partial differential equations/Dynamical systems
Non-existence of small-amplitude doubly periodic waves for dispersive equations
[Non-existence d'onde de petites amplitudes doublement périodiques pour les équations dispersives]

Présenté par : Jean-Michel Bony

David M. Ambrose1 ; J. Douglas Wright1
1 Department of Mathematics, Drexel University, 3141 Chestnut St., Philadelphia, PA 19104, USA
Comptes Rendus. Mathématique, Volume 352 (2014) no. 7-8, pp. 597-602.

Nous exprimons le problème d'existence de solutions périodiques en temps et en espace d'opérateurs d'évolution sous forme de problèmes de points fixes, pour certaines périodes de temps. Nous prouvons que, si une certaine estimation pour l'integrale de Duhamel existe, alors les solutions périodiques en temps ne peuvent être arbitrairement petites. Cela donne des résultats analogues pour le cas de la diffusion d'ondes périodiques dans l'espace sur la droite réelle, puisque la diffusion implique la non-existence d'onde de petites amplitudes. De plus, nos résultats viennent compléter les méthodes des petits diviseurs (comme par exemple la méthode de Craig–Wayne–Bourgain) pour prouver l'existence de solutions périodiques en temps de petites amplitudes (pour des frequences dans un certain ensemble).

We formulate the question of the existence of spatially periodic, time-periodic solutions for evolution equations as a fixed point problem, for certain temporal periods. We prove that if a certain estimate applies for the Duhamel integral, then time-periodic solutions cannot be arbitrarily small. This provides a partial analogue in the spatially periodic case of scattering results for dispersive equations on the real line, as scattering implies the non-existence of small-amplitude traveling waves. Furthermore, it also complements small-divisor methods (e.g., the Craig–Wayne–Bourgain method) for proving the existence of small-amplitude time-periodic solutions (again, for frequencies in certain set).

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2014.05.003
David M. Ambrose 1 ; J. Douglas Wright 1

1 Department of Mathematics, Drexel University, 3141 Chestnut St., Philadelphia, PA 19104, USA 
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David M. Ambrose; J. Douglas Wright. Non-existence of small-amplitude doubly periodic waves for dispersive equations. Comptes Rendus. Mathématique, Volume 352 (2014) no. 7-8, pp. 597-602. doi : 10.1016/j.crma.2014.05.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.05.003/

[1] D.M. Ambrose; J. Wilkening Global paths of time-periodic solutions of the Benjamin–Ono equation connecting pairs of traveling waves, Commun. Appl. Math. Comput. Sci., Volume 4 (2009), pp. 177-215

[2] D.M. Ambrose; J. Wilkening Computation of time-periodic solutions of the Benjamin–Ono equation, J. Nonlinear Sci., Volume 20 (2010) no. 3, pp. 277-308

[3] P. Baldi Periodic solutions of fully nonlinear autonomous equations of Benjamin–Ono type, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 30 (2013) no. 1, pp. 33-77

[4] J. Bourgain Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear PDE, Int. Math. Res. Not., Volume 1994 (1994) no. 11, p. 475 ff., approx. 21 pp. (electronic)

[5] J. Bourgain Construction of periodic solutions of nonlinear wave equations in higher dimension, Geom. Funct. Anal., Volume 5 (1995) no. 4, pp. 629-639

[6] F.M. Christ; M.I. Weinstein Dispersion of small amplitude solutions of the generalized Korteweg–de Vries equation, J. Funct. Anal., Volume 100 (1991) no. 1, pp. 87-109

[7] W. Craig; C.E. Wayne Newton's method and periodic solutions of nonlinear wave equations, Commun. Pure Appl. Math., Volume 46 (1993) no. 11, pp. 1409-1498

[8] R. de la Llave Variational methods for quasi-periodic solutions of partial differential equations, Pátzcuaro, 1998 (World Sci. Monogr. Ser. Math.), Volume vol. 6, World Sci. Publ., River Edge, NJ (2000), pp. 214-228

[9] B.A. Dubrovin A periodic problem for the Korteweg–de Vries equation in a class of short-range potentials, Funkcional. Anal. Priložen., Volume 9 (1975) no. 3, pp. 41-51

[10] W.-P. Düll Validity of the Korteweg–de Vries approximation for the two-dimensional water wave problem in the arc length formulation, Commun. Pure Appl. Math., Volume 65 (2012) no. 3, pp. 381-429

[11] M.B. Erdoğan; N. Tzirakis Global smoothing for the periodic KdV evolution, Int. Math. Res. Not., Volume 2013 (2013) no. 20, pp. 4589-4614

[12] G. Fibich; B. Ilan; G. Papanicolaou Self-focusing with fourth-order dispersion, SIAM J. Appl. Math., Volume 62 (2002) no. 4, pp. 1437-1462 (electronic)

[13] G. Fibich; B. Ilan; S. Schochet Critical exponents and collapse of nonlinear Schrödinger equations with anisotropic fourth-order dispersion, Nonlinearity, Volume 16 (2003) no. 5, pp. 1809-1821

[14] J. Ginibre; G. Velo Scattering theory in the energy space for a class of nonlinear Schrödinger equations, Trieste, 1984 (Pitman Res. Notes Math. Ser.), Volume vol. 141, Longman Sci. Tech., Harlow (1986), pp. 110-120

[15] G. Iooss; P.I. Plotnikov; J.F. Toland Standing waves on an infinitely deep perfect fluid under gravity, Arch. Ration. Mech. Anal., Volume 177 (2005) no. 3, pp. 367-478

[16] V.I. Karpman; A.G. Shagalov Stability of solitons described by nonlinear Schrödinger-type equations with higher-order dispersion, Physica D, Volume 144 (2000) no. 1–2, pp. 194-210

[17] C.E. Kenig; G. Ponce; L. Vega Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J., Volume 40 (1991) no. 1, pp. 33-69

[18] F. Linares; M. Scialom On the smoothing properties of solutions to the modified Korteweg–de Vries equation, J. Differ. Equ., Volume 106 (1993) no. 1, pp. 141-154

[19] Y. Liu Decay and scattering of small solutions of a generalized Boussinesq equation, J. Funct. Anal., Volume 147 (1997) no. 1, pp. 51-68

[20] Y. Matsuno New representations of multiperiodic and multisoliton solutions for a class of nonlocal soliton equations, J. Phys. Soc. Jpn., Volume 73 (2004) no. 12, pp. 3285-3293

[21] P.I. Plotnikov; J.F. Toland Nash–Moser theory for standing water waves, Arch. Ration. Mech. Anal., Volume 159 (2001) no. 1, pp. 1-83

[22] G. Ponce; L. Vega Nonlinear small data scattering for the generalized Korteweg–de Vries equation, J. Funct. Anal., Volume 90 (1990) no. 2, pp. 445-457

[23] W.A. Strauss Dispersion of low-energy waves for two conservative equations, Arch. Ration. Mech. Anal., Volume 55 (1974), pp. 86-92

[24] C.E. Wayne Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Commun. Math. Phys., Volume 127 (1990) no. 3, pp. 479-528

[25] J. Wilkening An infinite branching hierarchy of time-periodic solutions of the Benjamin–Ono equation, 2008 | arXiv

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