March  2017, 37(3): 1295-1321. doi: 10.3934/dcds.2017054

Modified energy functionals and the NLS approximation

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

Received  June 2016 Revised  November 2016 Published  December 2016

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].

Citation: Patrick Cummings, C. Eugene Wayne. Modified energy functionals and the NLS approximation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1295-1321. doi: 10.3934/dcds.2017054
References:
[1]

W. Craig, Nonstrictly hyperbolic nonlinear systems, Math. Ann., 277 (1987), 213-232. doi: 10.1007/BF01457361. Google Scholar

[2]

W. CraigC. 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. Google Scholar

[3]

W. -P. Düll, Justification of the Nonlinear Schrödinger approximation for a quasilinear wave equation, arXiv: 1602.08016Google Scholar

[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: 1605.08704Google Scholar

[5]

W.-P. DüllG. 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. Google Scholar

[6]

P. Germain, Space-time resonances, arXiv: 1102.1695Google Scholar

[7]

P. GermainN. 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. Google Scholar

[8]

J.K. HunterM. IfrimD. 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. Google Scholar

[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. Google Scholar

[10]

P. KirrmannG. 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. Google Scholar

[11]

D. Lannes, Space time resonances [after Germain, Masmoudi, Shatah], Séminaire Bourbaki. Vol. 2011/2012. Astérisque, 352 (2013), 355-388. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[15]

J. Shatah, Normal forms and quadratic nonlinear Klein-Gordon equations, Comm. Pure Appl. Math., 38 (1985), 685-696. doi: 10.1002/cpa.3160380516. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

show all references

References:
[1]

W. Craig, Nonstrictly hyperbolic nonlinear systems, Math. Ann., 277 (1987), 213-232. doi: 10.1007/BF01457361. Google Scholar

[2]

W. CraigC. 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. Google Scholar

[3]

W. -P. Düll, Justification of the Nonlinear Schrödinger approximation for a quasilinear wave equation, arXiv: 1602.08016Google Scholar

[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: 1605.08704Google Scholar

[5]

W.-P. DüllG. 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. Google Scholar

[6]

P. Germain, Space-time resonances, arXiv: 1102.1695Google Scholar

[7]

P. GermainN. 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. Google Scholar

[8]

J.K. HunterM. IfrimD. 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. Google Scholar

[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. Google Scholar

[10]

P. KirrmannG. 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. Google Scholar

[11]

D. Lannes, Space time resonances [after Germain, Masmoudi, Shatah], Séminaire Bourbaki. Vol. 2011/2012. Astérisque, 352 (2013), 355-388. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[15]

J. Shatah, Normal forms and quadratic nonlinear Klein-Gordon equations, Comm. Pure Appl. Math., 38 (1985), 685-696. doi: 10.1002/cpa.3160380516. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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