American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Existence of nontrivial solutions for equations of $p(x)$-Laplace type without Ambrosetti and Rabinowitz condition
  • PROC Home
  • This Issue
  • Next Article
    Stability of interacting traveling waves in reaction-convection-diffusion systems
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

[1]

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
[2]

Rémi Carles, Christof Sparber. Semiclassical wave packet dynamics in Schrödinger equations with periodic potentials. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 759-774. doi: 10.3934/dcdsb.2012.17.759
[3]

Jaime Angulo Pava, Nataliia Goloshchapova. On the orbital instability of excited states for the NLS equation with the δ-interaction on a star graph. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5039-5066. doi: 10.3934/dcds.2018221
[4]

Alexandre N. Carvalho, Jan W. Cholewa. NLS-like equations in bounded domains: Parabolic approximation procedure. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 57-77. doi: 10.3934/dcdsb.2018005
[5]

Jong-Shenq Guo, Hirokazu Ninomiya, Chin-Chin Wu. Existence of a rotating wave pattern in a disk for a wave front interaction model. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1049-1063. doi: 10.3934/cpaa.2013.12.1049
[6]

Salim Meddahi, David Mora. Nonconforming mixed finite element approximation of a fluid-structure interaction spectral problem. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 269-287. doi: 10.3934/dcdss.2016.9.269
[7]

George Avalos, Thomas J. Clark. A mixed variational formulation for the wellposedness and numerical approximation of a PDE model arising in a 3-D fluid-structure interaction. Evolution Equations & Control Theory, 2014, 3 (4) : 557-578. doi: 10.3934/eect.2014.3.557
[8]

Santiago Montaner, Arnaud Münch. Approximation of controls for linear wave equations: A first order mixed formulation. Mathematical Control & Related Fields, 2019, 9 (4) : 729-758. doi: 10.3934/mcrf.2019030
[9]

Remi Sentis. Models and simulations for the laser-plasma interaction and the three-wave coupling problem. Discrete & Continuous Dynamical Systems - S, 2012, 5 (2) : 329-343. doi: 10.3934/dcdss.2012.5.329
[10]

Belkacem Said-Houari, Flávio A. Falcão Nascimento. Global existence and nonexistence for the viscoelastic wave equation with nonlinear boundary damping-source interaction. Communications on Pure & Applied Analysis, 2013, 12 (1) : 375-403. doi: 10.3934/cpaa.2013.12.375
[11]

Ademir Pastor. On three-wave interaction Schrödinger systems with quadratic nonlinearities: Global well-posedness and standing waves. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2217-2242. doi: 10.3934/cpaa.2019100
[12]

Rémi Carles, Erwan Faou. Energy cascades for NLS on the torus. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2063-2077. doi: 10.3934/dcds.2012.32.2063
[13]

Wouter Rogiest, Koen De Turck, Koenraad Laevens, Dieter Fiems, Sabine Wittevrongel, Herwig Bruneel. On the optimality of packet-oriented scheduling in photonic switches with delay lines. Numerical Algebra, Control & Optimization, 2011, 1 (4) : 727-747. doi: 10.3934/naco.2011.1.727
[14]

Tetsuji Hirayama. Analysis of multiclass feedback queues and its application to a packet scheduling problem. Journal of Industrial & Management Optimization, 2010, 6 (3) : 541-568. doi: 10.3934/jimo.2010.6.541
[15]

Vassilis Rothos. Subharmonic bifurcations of localized solutions of a discrete NLS equation. Conference Publications, 2005, 2005 (Special) : 756-767. doi: 10.3934/proc.2005.2005.756
[16]

A. Adam Azzam. Scattering for the two dimensional NLS with (full) exponential nonlinearity. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1071-1101. doi: 10.3934/cpaa.2018052
[17]

Casey Jao. Energy-critical NLS with potentials of quadratic growth. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 563-587. doi: 10.3934/dcds.2018025
[18]

Kai Yang. The focusing NLS on exterior domains in three dimensions. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2269-2297. doi: 10.3934/cpaa.2017112
[19]

Scipio Cuccagna. Orbitally but not asymptotically stable ground states for the discrete NLS. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 105-134. doi: 10.3934/dcds.2010.26.105
[20]

Panayotis Panayotaros. Continuation and bifurcations of breathers in a finite discrete NLS equation. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1227-1245. doi: 10.3934/dcdss.2011.4.1227

 Impact Factor: 

Tools

Metrics

  • PDF downloads (37)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences