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2015, 2015(special): 267-275. doi: 10.3934/proc.2015.0267

Interaction of oscillatory packets of water waves

Martina Chirilus-Bruckner 1, and  Guido Schneider 2,

1. 

Mathematical Institute, University of Leiden, 2300 RA Leiden, Netherlands
2. 

Institut für Analysis, Dynamik und Modellierung, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart

Received  September 2014 Revised  March 2015 Published  November 2015

For surface gravity water waves we give a detailed analysis of the interaction of two NLS described wave packets with different carrier waves. We separate the internal dynamics of each wave packet from the dynamics caused by the interaction and prove the validity of a formula for the envelope shift caused by the interaction of the wave packets.
Keywords: wave packet interaction., NLS approximation.
Mathematics Subject Classification: Primary: 76B15; Secondary: 35Q55, 35A3.
Citation: Martina Chirilus-Bruckner, Guido Schneider. Interaction of oscillatory packets of water waves. Conference Publications, 2015, 2015 (special) : 267-275. doi: 10.3934/proc.2015.0267
References:
[1]

A. Babin and A. Figotin., Linear superposition in nonlinear wave dynamics., Rev. Math. Phys., 18 (2006), 971.   Google Scholar

[2]

M. Chirilus-Bruckner, G. Schneider and H. Uecker., On the interaction of NLS-described modulating pulses with different carrier waves., Math. Methods Appl. Sci., 30 (2007), 1965.   Google Scholar

[3]

M. Chirilus-Bruckner, C. Chong, G. Schneider and H. Uecker., Separation of internal and interaction dynamics for NLS-described wave packets with different carrier waves., J. Math. Anal. Appl., 347 (2008), 304.   Google Scholar

[4]

M. Chirilus-Bruckner and G. Schneider., Detection of standing pulses in periodic media by pulse interaction., J. Differential Equations, 253 (2012), 2161.   Google Scholar

[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., Archive for Rational Mechanics and Analysis, (2015).   Google Scholar

[6]

M. Oikawa and N. Yajima., A perturbation approach to nonlinear systems. ii. interaction of nonlinear modulated waves., Journal of the Physical Society of Japan, 37 (1974), 486.   Google Scholar

[7]

R.D. Pierce and C.E. Wayne., On the validity of mean-field amplitude equations for counterpropagating wavetrains., Nonlinearity, 8 (1995), 769.   Google Scholar

[8]

G. Schneider, H. Uecker and M. Wand., Interaction of modulated pulses in nonlinear oscillator chains., Journal of Difference Eq. and Appl., 17 (2011), 279.   Google Scholar

[9]

L. Tkeshelashvili, S. Pereira and K. Busch., General theory of nonresonant wave interaction: Giant soliton shift in photonic band gap materials., Europhysics Letters, 68 (2004).   Google Scholar

[10]

N. Totz and S. Wu., A rigorous justification of the modulation approximation to the 2D full water wave problem., Commun. Math. Phys., 310 (2012), 817.   Google Scholar

[11]

V.E. Zakharov., Stability of periodic waves of finite amplitude on the surface of a deep fluid., Sov. Phys. J. Appl. Mech. Tech. Phys, 4 (1968), 190.   Google Scholar

show all references

References:
[1]

A. Babin and A. Figotin., Linear superposition in nonlinear wave dynamics., Rev. Math. Phys., 18 (2006), 971.   Google Scholar

[2]

M. Chirilus-Bruckner, G. Schneider and H. Uecker., On the interaction of NLS-described modulating pulses with different carrier waves., Math. Methods Appl. Sci., 30 (2007), 1965.   Google Scholar

[3]

M. Chirilus-Bruckner, C. Chong, G. Schneider and H. Uecker., Separation of internal and interaction dynamics for NLS-described wave packets with different carrier waves., J. Math. Anal. Appl., 347 (2008), 304.   Google Scholar

[4]

M. Chirilus-Bruckner and G. Schneider., Detection of standing pulses in periodic media by pulse interaction., J. Differential Equations, 253 (2012), 2161.   Google Scholar

[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., Archive for Rational Mechanics and Analysis, (2015).   Google Scholar

[6]

M. Oikawa and N. Yajima., A perturbation approach to nonlinear systems. ii. interaction of nonlinear modulated waves., Journal of the Physical Society of Japan, 37 (1974), 486.   Google Scholar

[7]

R.D. Pierce and C.E. Wayne., On the validity of mean-field amplitude equations for counterpropagating wavetrains., Nonlinearity, 8 (1995), 769.   Google Scholar

[8]

G. Schneider, H. Uecker and M. Wand., Interaction of modulated pulses in nonlinear oscillator chains., Journal of Difference Eq. and Appl., 17 (2011), 279.   Google Scholar

[9]

L. Tkeshelashvili, S. Pereira and K. Busch., General theory of nonresonant wave interaction: Giant soliton shift in photonic band gap materials., Europhysics Letters, 68 (2004).   Google Scholar

[10]

N. Totz and S. Wu., A rigorous justification of the modulation approximation to the 2D full water wave problem., Commun. Math. Phys., 310 (2012), 817.   Google Scholar

[11]

V.E. Zakharov., Stability of periodic waves of finite amplitude on the surface of a deep fluid., Sov. Phys. J. Appl. Mech. Tech. Phys, 4 (1968), 190.   Google Scholar

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