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On the Cauchy problem for the Zakharov-Rubenchik/ Benney-Roskes system

Hung Luong, Norbert J. Mauser, Jean-Claude Saut
2018 Communications on Pure and Applied Analysis  
We address various issues concerning the Cauchy problem for the Zakharov-Rubenchik system(known as the Benney-Roskes system in water waves theory), which models the interaction of short and long waves  ...  Motivated by the transverse stability/instability of the one-dimensional solitary wave (line solitary), we study the Cauchy problem in the background of a line solitary wave. 2000 Mathematics Subject Classification  ...  We thank the anonymous referees for valuable comments and criticisms that helped us to improve the manuscript.  ... 
doi:10.3934/cpaa.2018075 fatcat:7y5lco7he5cd7a7axmywvn6bbu

On the Cauchy problem for the Zakharov-Rubenchik/Benney-Roskes system [article]

Hung Luong, Norbert Mauser, Jean-Claude Saut
2018 arXiv   pre-print
We address various issues concerning the Cauchy problem for the Zakharov-Rubenchik system (known as the Benney-Roskes system in water waves theory), which models the interaction of short and long waves  ...  Motivated by the transverse stability/instability of the one-dimensional solitary wave (line solitary), we study the Cauchy problem in the background of a line solitary wave.  ...  We acknowledge an anonymous referee for valuable comments and criticisms that helped us to improve the manuscript.  ... 
arXiv:1802.06586v1 fatcat:mwmgqyyvdbdn5gao6tfdsvmnl4

Arbitrarily high-order energy-preserving schemes for the Zakharov-Rubenchik equation [article]

Gengen Zhang and Chaolong Jiang and Hao Huang
2022 arXiv   pre-print
In this paper, we present a high-order energy-preserving scheme for solving Zakharov-Rubenchik equation.  ...  In addition, an efficient fixed-point iteration is proposed to solve the resulting nonlinear equations of the proposed schemes. Several experiments are provided to validate the theoretical results.  ...  Acknowledgment The work is supported by the National Natural Science Foundation of China (Grant Nos.11701110, 11901513), and the Yunnan Fundamental Research Projects (Grant Nos. 202101AT070208, 202101AS070044  ... 
arXiv:2205.12154v1 fatcat:ic7cbdbyabhojbdxexs3zu4mcy

High precision numerical approach for the Davey-Stewartson II equation for Schwartz class initial data [article]

C. Klein, K. McLaughlin, N. Stoilov
2019 arXiv   pre-print
The code can treat integrable and non-integrable DS II equations. Moreover, it has the same existing codes for DS II.  ...  Since the DS equation is a nonlocal, nonlinear Schrödinger equation with a singular symbol for the nonlocality, standard Fourier methods in practice only reach accuracies of the order of 10^-6 or less  ...  However the Zakharov-Rubenchik systems ZR [40] make physical sense also in three spatial dimensions.  ... 
arXiv:1911.11721v1 fatcat:hl4m52pfh5dspjpnhatqttr2nu

IST versus PDE, a comparative study [article]

C. Klein, J.-C. Saut
2014 arXiv   pre-print
We survey and compare, mainly in the two-dimensional case, various results obtained by IST and PDE techniques for integrable equations.  ...  We also comment on what can be predicted from integrable equations on non integrable ones.  ...  short and long waves, the Benney-Roskes, Zakharov-Rubenchik systems ( [21, 229] ).  ... 
arXiv:1409.2020v2 fatcat:iftou2gru5bypezo4w36l34oza

Transverse nonlinear instability of solitary waves for some Hamiltonian PDE's

Frederic Rousset, Nikolay Tzvetkov
2008 Journal des Mathématiques Pures et Appliquées  
Our result applies to the generalized KP-I equation, the Nonlinear Schrödinger equation, the generalized Boussinesq system and the Zakharov-Kuznetsov equation and we hope that it may be useful in other  ...  Notre résultat s'applique aux équations de KP-I généralisées, Schrödinger non linéaire et Zakharov-Kuzntetsov et à un système de Boussinesq et on espère qu'il pourra être utile dans d'autres contextes.  ...  Acknowledgement We are indebted to Jean-Claude Saut for several discussions on the subject and in particular for providing us Ref. [13] .  ... 
doi:10.1016/j.matpur.2008.07.004 fatcat:nugrmkl3mnfczaovqtnezlceay

Transverse nonlinear instability of solitary waves for some Hamiltonian PDE's [article]

Frederic Rousset, Nikolay Tzvetkov
2008 arXiv   pre-print
Our result applies to the generalized KP-I equation, the Nonlinear Schr\"odinger equation, the generalized Boussinesq system and the Zakharov-Kuznetsov equation and we hope that it may be useful in other  ...  We present a general result of transverse nonlinear instability of 1-d solitary waves for Hamiltonian PDE's for both periodic or localized transverse perturbations.  ...  We are indebted to Jean-Claude Saut for several discussions on the subject and in particular for providing us the reference [13] .  ... 
arXiv:0804.1308v1 fatcat:fynoztvg2vbyxd3qkaah4jt3mu

(In)stability and asymptotic analysis of the Zakharov-Rubenchik system and related wave type equations

Hung Luong
2018 unpublished
We consider these models as the asymptotic models of the very general "water wave equation system" like the Zakharov-Rubenchik system.  ...  In the Chapter 3, we study the Zakharov-Rubenchik/Benney-Roskes (Z-R/B-R) system.  ...  Mauser for introducing me to the problem and many helpful discussions. I also thank the anonymous referees for valuable comments and criticisms that helped me to improve the manuscript.  ... 
doi:10.25365/thesis.53733 fatcat:4p42f66efzeqnkgw67zx4hqfta

Bifurcations of solitons and their stability

E.A. Kuznetsov, F. Dias
2011 Physics reports  
These solitons essentially differed from those in shallow water where the KDV equation could be used.  ...  Two approaches are used for the soliton stability analysis.  ...  Acknowledgments The authors thank V.E. Zakharov  ... 
doi:10.1016/j.physrep.2011.06.002 fatcat:etj423qoh5ghpotfjfc24uw4r4

Quantum fluids of light

Iacopo Carusotto, Cristiano Ciuti
2013 Reviews of Modern Physics  
out the rich phenomenology offered by the implementation of artificial gauge fields for photons.  ...  In particular, we present different mechanisms to obtain efficient photon blockade, we discuss the novel quantum phases that are expected to appear in arrays of strongly nonlinear cavities, and we point  ...  Acknowledgments This review article could never be completed without the continuous stimulating discussions with the many friends and colleagues with whom we had the chance of collaborating during all  ... 
doi:10.1103/revmodphys.85.299 fatcat:pv7q36ig3zeexgn6ppqpsithbe

Solitons in nonlinear lattices

Yaroslav V. Kartashov, Boris A. Malomed, Lluis Torner
2011 Reviews of Modern Physics  
In both the 1D and 2D cases, the mechanism which creates solitons in NLs is principally different from its counterpart in linear lattices, as the solitons are created directly, rather than bifurcating  ...  In multidimensional settings, the stability of solitons supported by the spatial modulation of the nonlinearity is a truly challenging problem, for the theoretical and experimental studies alike.  ...  ACKNOWLEDGMENTS We thank all our colleagues and collaborators for fruitful interactions over the past few years, in conducting various research projects whose results are reported, inter alia, here, and  ... 
doi:10.1103/revmodphys.83.247 fatcat:wpgr7nvrrjacvirgqqxczhwuca

Optics in the relativistic regime

Gerard A. Mourou, Toshiki Tajima, Sergei V. Bulanov
2006 Reviews of Modern Physics  
The laser fields can be so intense as to make the accelerating field large enough for general relativistic effects ͑via the equivalence principle͒ to be examined in the laboratory.  ...  The coupling of an intense laser field to matter also has implications for the study of the highest energies in astrophysics, such as ultrahigh-energy cosmic rays, with energies in excess of 10 20 eV.  ...  than in 1D and 2D because wave collapse in 3D configurations results in the development of a 3D singularity ͑Zakharov, 1972; Kuznetsov, Rubenchik, and Zakharov, 1986; Kuznetsov, 1996͒.  ... 
doi:10.1103/revmodphys.78.309 fatcat:4vyljdmjp5cdzhajs3a5b5zb3m

Ultrashort filaments of light in weakly ionized, optically transparent media

L Bergé, S Skupin, R Nuter, J Kasparian, J-P Wolf
2007 Reports on progress in physics (Print)  
Particular emphasis is laid to the theoretical modeling of the propagation equations, whose physical ingredients are discussed from numerical simulations.  ...  Differences between femtosecond pulses propagating in gaseous or condensed materials are underlined.  ...  Acknowledgments The Teramobile project is funded by the Centre National de la Recherche Scientifique (CNRS, France) and the Deutsche Forschungsgemeinschaft (DFG, Germany), with contribution by the French  ... 
doi:10.1088/0034-4885/70/10/r03 fatcat:udxbesy5o5dk7kde7kaxu2v4ni