Modified energy functionals and the NLS approximation

Patrick Cummings; C. Eugene Wayne

doi:10.3934/dcds.2017054

Article Contents

2017, Volume 37, Issue 3: 1295-1321. Doi: 10.3934/dcds.2017054

This issue Previous Article A remark on the partial regularity of a suitable weak solution to the Navier-Stokes Cauchy problem Next Article High-frequency instabilities of small-amplitude solutions of Hamiltonian PDEs

Modified energy functionals and the NLS approximation

Patrick Cummings and
C. Eugene Wayne

Department of Mathematics and Statistics, Boston University, Boston, MA 02215, USA

Received: May 31, 2016

Revised: October 31, 2016

Published: May 2017

Abstract Full Text(HTML) Figure(1) Related Papers Cited by

Abstract

We consider a model equation from [14] that captures important properties of the water wave equation. We give a new proof of the fact that wave packet solutions of this equation are approximated by the nonlinear Schrödinger equation. This proof both simplifies and strengthens the results of [14] so that the approximation holds for the full interval of existence of the approximate NLS solution rather than just a subinterval. Furthermore, the proof avoids the problems associated with inverting the normal form transform in [14] by working with a modified energy functional motivated by [1] and [8].

Keywords:

Mathematics Subject Classification: 35L72, 35Q55, 70K30, 70K45, 35Q35.

Citation:

Full Text(HTML)

Figure 1. Partition of k$\ell$ -plane.

Download: Full-size image PowerPoint slide

Related Papers

Cited by

References

[1]	W. Craig, Nonstrictly hyperbolic nonlinear systems, Math. Ann., 277 (1987), 213-232. doi: 10.1007/BF01457361.
[2]	W. Craig, C. Sulem and P.-L. Sulem, Nonlinear modulation of gravity waves: A rigorous approach, Nonlinearity, 5 (1992), 497-522. doi: 10.1088/0951-7715/5/2/009.
[3]	W. -P. Düll, Justification of the Nonlinear Schrödinger approximation for a quasilinear wave equation, arXiv: 1602.08016
[4]	W. -P. Düll and M. Heẞ, Existence of long time solutions and validity of the Nonlinear Schrödinger approximation for a quasilinear dispersive equation, arXiv: 1605.08704
[5]	W.-P. Düll, G. Schneider and C.E. Wayne, Justification of the nonlinear Schrödinger equation for the evolution of gravity driven 2D surface water waves in a canal of finite depth, Arch. Ration. Mech. Anal., 220 (2016), 543-602. doi: 10.1007/s00205-015-0937-z.
[6]	P. Germain, Space-time resonances, arXiv: 1102.1695
[7]	P. Germain, N. Masmoudi and J. Shatah, Global solutions for the gravity water waves equation in dimension 3, Ann. of Math., 175 (2012), 691-754. doi: 10.4007/annals.2012.175.2.6.
[8]	J.K. Hunter, M. Ifrim, D. Tataru and T.K. Wong, Long time solutions for a Burgers-Hilbert equation via a modified energy method, Proc. Amer. Math. Soc., 143 (2015), 3407-3412. doi: 10.1090/proc/12215.
[9]	L.A. Kalyakin, Asymptotic decay of a one-dimensional wave packet in a nonlinear dispersive medium, Mat. Sb. (N.S.), 132 (1987), 470-495, 592.
[10]	P. Kirrmann, G. Schneider and A. Mielke, The validity of modulation equations for extended systems with cubic nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A, 122 (1992), 85-91. doi: 10.1017/S0308210500020989.
[11]	D. Lannes, Space time resonances [after Germain, Masmoudi, Shatah], Séminaire Bourbaki. Vol. 2011/2012. Astérisque, 352 (2013), 355-388.
[12]	G. Schneider, Justification of modulation equations for hyperbolic systems via normal forms, NoDEA Nonlinear Differential Equations Appl., 5 (1998), 69-82. doi: 10.1007/s000300050034.
[13]	G. Schneider, Justification and failure of the nonlinear Schröodinger equation in case of non-trivial quadratic resonances, J. Differential Equations, 216 (2005), 354-386. doi: 10.1016/j.jde.2005.04.018.
[14]	G. Schneider and C.E. Wayne, Justification of the NLS approximation for a quasilinear water wave model, J. Differential Equations, 251 (2011), 238-269. doi: 10.1016/j.jde.2011.04.011.
[15]	J. Shatah, Normal forms and quadratic nonlinear Klein-Gordon equations, Comm. Pure Appl. Math., 38 (1985), 685-696. doi: 10.1002/cpa.3160380516.
[16]	N. Totz, A justification of the modulation approximation to the 3D full water wave problem, Comm. Math. Phys., 335 (2015), 369-443. doi: 10.1007/s00220-014-2259-7.
[17]	N. Totz and S. Wu, A rigorous justification of the modulation approximation to the 2D full water wave problem, Comm. Math. Phys., 310 (2012), 817-883. doi: 10.1007/s00220-012-1422-2.
[18]	V.E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, Journal of Applied Mechanics and Technical Physics, 9 (1968), 190-194. doi: 10.1007/BF00913182.