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

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  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].

Keywords: Modulation equation, nonlinear Schrödinger equation, normal forms, modified energy, resonance.
Mathematics Subject Classification: 35L72, 35Q55, 70K30, 70K45, 35Q35.
Citation: Patrick Cummings, C. Eugene Wayne. Modified energy functionals and the NLS approximation. Discrete and Continuous Dynamical Systems, 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.

[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.

show all references

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.

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