# American Institute of Mathematical Sciences

July  2018, 17(4): 1573-1594. doi: 10.3934/cpaa.2018075

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

 1 Fak. Mathematik, University of Vienna, Oskar MorgensternPlatz 1, A-1090 Wien, Austria 2 Wolfgang Pauli Institute c/o Fak. Math. Univ. Vienna, Oskar MorgensternPlatz 1, A-1090 Wien, Austria 3 Laboratoire de Mathématiques, UMR 8628, Université Paris-Saclay, Paris-Sud and CNRS, F-91405 Orsay, France

Received  March 2017 Revised  February 2018 Published  April 2018

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 in many physical situations. 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.

Citation: Hung Luong, Norbert J. Mauser, Jean-Claude Saut. On the Cauchy problem for the Zakharov-Rubenchik/ Benney-Roskes system. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1573-1594. doi: 10.3934/cpaa.2018075
##### References:
 [1] M. J. Ablowitz and H. Segur, On the evolution of packets of water waves, J. Fluid Mech., 92 (1979), 691-715. Google Scholar [2] H. Added and S. Added, Equations of Langmuir turbulence and nonlinear Schrödinger equation: smoothness and approximation, J. Funct. Anal., 76 (1988), 183-210. Google Scholar [3] H. Added and S. Added, Existence globale de solutions fortes pour les équations de la turbulence de Langmuir en dimension 2, C. R. Acad. Sci. Paris Sér. I Math., 299 (1984), 551-554. Google Scholar [4] I. Bejenaru, S. Herr, J. Holmer and D. Tataru, On the 2D Zakharov system with L2-Schrödinger data, Nonlinearity, 22 (2009), 1063-1089. Google Scholar [5] D. J. Benney and A. C. Newell, The propagation of nonlinear envelopes, J. Math. and Phys., 46 (1967), 133-139. Google Scholar [6] D. J. Benney and G. J. Roskes, Waves instabilities, Stud. Appl. Math., 48 (1969), 377-385. Google Scholar [7] J. Bourgain, On the Cauchy and invariant measure problem for the periodic Zakharov system, Duke Math. J., 76 (1994), 175-202. Google Scholar [8] J. Bourgain and J. Colliander, On wellposedness of the Zakharov system, IRMN, 11 (1996), 515-546. Google Scholar [9] R. Cipolatti, On the existence of standing waves for a Davey-Stewartson system, Comm. Partial Differential Equations, 17 (1992), 967-988. Google Scholar [10] R. Cipolatti, On the instability of ground states for a Davey-Stewartson system, Ann.Inst. Poincaré H., Phys.Théor., 58 (1993), 85-104. Google Scholar [11] T. Colin, Rigorous derivation of the nonlinear Schrödinger equation and Davey-Stewartson systems from quadratic hyperbolic systems, Asymptotic Analysis, 31 (2002), 69-91. Google Scholar [12] T. Colin and D. Lannes, Justification of and long-wave correction to Davey-Stewartson systems from quadratic hyperbolic systems, Disc. Cont. Dyn. Systems, 11 (2004), 83-100. Google Scholar [13] J.C. Cordero Ceballos, Supersonic limit for the Zakharov-Rubenchik system, J. Diff. Eq., 261 (2016), 5260-5288. Google Scholar [14] A. Davey and K. Stewartson, One three-dimensional packets of water waves, Proc. Roy. Soc. Lond. A, 338 (1974), 101-110. Google Scholar [15] V. D. Djordjevic and L. G. Redekopp, On two-dimensional packets of capillary-gravity waves, J. Fluid Mech., 79 (1977), 703-714. Google Scholar [16] J.-M. Ghidaglia and J.-C. Saut, On the initial value problem for the Davey-Stewartson systems, Nonlinearity, 3 (1990), 475-506. Google Scholar [17] J.-M. Ghidaglia and J.-C. Saut, Non existence of traveling wave solutions to nonelliptic nonlinear Schrödinger equations, J. Nonlinear Sci., 6 (1996), 139-145. Google Scholar [18] J. Ginibre, Y. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system, J. Funct. Analysis, 151 (1997), 384-436. Google Scholar [19] L. Glangetas and F. Merle, Existence of self-similar blow-up solutions for Zakharov equation in dimension two. Ⅰ, Comm. Math. Phys., 160 (1994), 173-215. Google Scholar [20] L. Glangetas and F. Merle, Existence of self-similar blow-up solutions for Zakharov equation in dimension two. Ⅱ, Comm. Math. Phys., 160 (1994), 349-389. Google Scholar [21] T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., XLI (1988), 891-907. Google Scholar [22] C. Kenig, G. Ponce and L. Vega, On the Zakharov and Zakharov-Schulman systems, J. Funct; Anal., 127 (1995), 204-234. Google Scholar [23] S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math., 34 (1981), 481-524. Google Scholar [24] D. Lannes, Sharp estimates for pseudo-differential operators with symbols of limited smoothness and commutators, J. Funct. Anal., 232 (2006), 495-539. Google Scholar [25] D. Lannes, Water Waves: Mathematical Theory and Asymptotics, Mathematical Surveys and Monographs, vol 188 (2013), AMS, Providence.Google Scholar [26] F. Linares and C. Matheus, Well-posedness for the 1-D Zakharov system, Advances in Diff. Equations, 14 (2009), 261-288. Google Scholar [27] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod; Gauthier-Villars, Paris, 1989.Google Scholar [28] A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences volume 53, Springer-Verlag, New-York, 1984.Google Scholar [29] T. Mizumachi, Stability of line solitons for the KP-Ⅱ equation in ${{\mathbb{R}}^{2}}$, Memoirs of the AMS, vol. 238, number 1125, (2015).Google Scholar [30] T. Mizumachi and N. Tzvetkov, Stability of the line soliton of the KP-Ⅱ equation under periodic transverse perturbations, Math. Ann., 352 (2012), 659-690. Google Scholar [31] C. Obrecht, Thèse de Doctorat, Université Paris-Sud (2015) and article in preparation.Google Scholar [32] M. Ohta, Stability and instability of standing waves for the generalized Davey-Stewartson system, Diff. Int. Eq., 8 (1995), 1775-1788. Google Scholar [33] M. Ohta, Instability of standing waves for the generalized Davey-Stewartson system, Ann. Inst. Poincaré H., Phys. Théor., 62 (1995), 69-80. Google Scholar [34] M. Ohta, Blow-up solutions and strong instability of standing waves for the generalized DaveyStewartson system, Ann. Inst. Poincaré H., Phys. Théor., 63 (1995), 111-117. Google Scholar [35] F. Oliveira, Stability of the solitons for the one-dimensional Zakharov-Rubenchik system, Physica D, 175 (2003), 220-240. Google Scholar [36] F. Oliveira, Adiabatic limit of the Zakharov-Rubenchik system, Reports on Mathematical Physics, 61 (2008), 13-27. Google Scholar [37] T. Ozawa and Y. Tsutsumi, Existence and smoothing effect of solutions for the Zakharov equations, Publ. Res. Inst. Math. Sci., Kyoto University. Research Institute for Mathematical Sciences. Publications, 28 (1992), 329-361. Google Scholar [38] T. Ozawa and Y. Tsutsumi, The nonlinear Schrödinger limit and the initial layer of the Zakharov equations, Proc. Jap. Acad. A, 67 (1991), 113-116. Google Scholar [39] T. Ozawa and Y. Tsutsumi, The nonlinear Schrödinger limit and the initial layer of the Zakharov equations, Diff. Int. Eq., 5 (1992), 721-745. Google Scholar [40] T. Passot, P.-L. Sulem and C. Sulem, Generalization of acoustic fronts by focusing ave packets, Physica D, 94 (1996), 168-187. Google Scholar [41] G. Ponce and J.-C. Saut, Well-posedness for the Benney-Roskes-Zakharov-Rubenchik system, Discrete Cont. Dynamical Systems, 13 (2005), 811-825. Google Scholar [42] F. Rousset and N. Tzvetkov, Transverse instability of the line solitary water-waves, Invent. Math., 184 (2011), 257-388. Google Scholar [43] F. Rousset and N. Tzvetkov, Transverse nonlinear instability for two-dimensional dispersive models, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 477-496. Google Scholar [44] F. Rousset and N. Tzvetkov, Transverse nonlinear instability of solitary waves for some Hamiltonian PDE's, J. Math. Pures et Appl., 80 (2008), 550-590. Google Scholar [45] S. H. Schochet and M.I. Weinstein, The nonlinear Schrödinger limit of the Zakharov governing Langmuir turbulence, Comm. Math. Phys., 106 (1986), 569-580. Google Scholar [46] C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation, Springer-Verlag, Applied Mathematical Sciences 139 New York, Berlin, 1999.Google Scholar [47] C. Sulem and P.-L. Sulem, Quelques résultats de régularité pour les équations de la turbulence de Langmuir, C.R. Ac. Sci. Paris Sér. A-B, 289 (1979), A173-A176. Google Scholar [48] H. Takaoka, Well-posedness for the Zakharov system with the periodic boundary condition, Diff. and Int. equations, 6 (1999), 789-810. Google Scholar [49] N. Tzvetkov, Low regularity solutions for a generalized Zakharov system, Diff. and Int. Equations, 13 (2000), 423-440. Google Scholar [50] V. E. Zakharov, Collapse of Langmuir waves, Sov. Phys. JETP, 35 (1972), 908-914. Google Scholar [51] V. E. Zakharov, Weakly nonlinear waves on the surface of an ideal finite depth fluid, Amer. Math. Soc. Transl., 182 (1998), 167-197. Google Scholar [52] V. E. Zakharov and E. A. Kuznetsov, Hamiltonian formalism for nonlinear waves, PhysicsUspekhi, 40 (1997), 1087-1116. Google Scholar [53] V. E. Zakharov and A. M. Rubenchik, Nonlinear interaction of high-frequency and low frequency waves, Prikl. Mat. Techn. Phys., 5 (1972), 84-98. Google Scholar [54] Xiaofei Zhao and Ziyi Li, Numerical methods and simulations for the dynamics of onedimensional Zakharov-Rubenchik equations, J. Sci. Comput., 59 (2014), 412-438. Google Scholar

show all references

##### References:
 [1] M. J. Ablowitz and H. Segur, On the evolution of packets of water waves, J. Fluid Mech., 92 (1979), 691-715. Google Scholar [2] H. Added and S. Added, Equations of Langmuir turbulence and nonlinear Schrödinger equation: smoothness and approximation, J. Funct. Anal., 76 (1988), 183-210. Google Scholar [3] H. Added and S. Added, Existence globale de solutions fortes pour les équations de la turbulence de Langmuir en dimension 2, C. R. Acad. Sci. Paris Sér. I Math., 299 (1984), 551-554. Google Scholar [4] I. Bejenaru, S. Herr, J. Holmer and D. Tataru, On the 2D Zakharov system with L2-Schrödinger data, Nonlinearity, 22 (2009), 1063-1089. Google Scholar [5] D. J. Benney and A. C. Newell, The propagation of nonlinear envelopes, J. Math. and Phys., 46 (1967), 133-139. Google Scholar [6] D. J. Benney and G. J. Roskes, Waves instabilities, Stud. Appl. Math., 48 (1969), 377-385. Google Scholar [7] J. Bourgain, On the Cauchy and invariant measure problem for the periodic Zakharov system, Duke Math. J., 76 (1994), 175-202. Google Scholar [8] J. Bourgain and J. Colliander, On wellposedness of the Zakharov system, IRMN, 11 (1996), 515-546. Google Scholar [9] R. Cipolatti, On the existence of standing waves for a Davey-Stewartson system, Comm. Partial Differential Equations, 17 (1992), 967-988. Google Scholar [10] R. Cipolatti, On the instability of ground states for a Davey-Stewartson system, Ann.Inst. Poincaré H., Phys.Théor., 58 (1993), 85-104. Google Scholar [11] T. Colin, Rigorous derivation of the nonlinear Schrödinger equation and Davey-Stewartson systems from quadratic hyperbolic systems, Asymptotic Analysis, 31 (2002), 69-91. Google Scholar [12] T. Colin and D. Lannes, Justification of and long-wave correction to Davey-Stewartson systems from quadratic hyperbolic systems, Disc. Cont. Dyn. Systems, 11 (2004), 83-100. Google Scholar [13] J.C. Cordero Ceballos, Supersonic limit for the Zakharov-Rubenchik system, J. Diff. Eq., 261 (2016), 5260-5288. Google Scholar [14] A. Davey and K. Stewartson, One three-dimensional packets of water waves, Proc. Roy. Soc. Lond. A, 338 (1974), 101-110. Google Scholar [15] V. D. Djordjevic and L. G. Redekopp, On two-dimensional packets of capillary-gravity waves, J. Fluid Mech., 79 (1977), 703-714. Google Scholar [16] J.-M. Ghidaglia and J.-C. Saut, On the initial value problem for the Davey-Stewartson systems, Nonlinearity, 3 (1990), 475-506. Google Scholar [17] J.-M. Ghidaglia and J.-C. Saut, Non existence of traveling wave solutions to nonelliptic nonlinear Schrödinger equations, J. Nonlinear Sci., 6 (1996), 139-145. Google Scholar [18] J. Ginibre, Y. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system, J. Funct. Analysis, 151 (1997), 384-436. Google Scholar [19] L. Glangetas and F. Merle, Existence of self-similar blow-up solutions for Zakharov equation in dimension two. Ⅰ, Comm. Math. Phys., 160 (1994), 173-215. Google Scholar [20] L. Glangetas and F. Merle, Existence of self-similar blow-up solutions for Zakharov equation in dimension two. Ⅱ, Comm. Math. Phys., 160 (1994), 349-389. Google Scholar [21] T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., XLI (1988), 891-907. Google Scholar [22] C. Kenig, G. Ponce and L. Vega, On the Zakharov and Zakharov-Schulman systems, J. Funct; Anal., 127 (1995), 204-234. Google Scholar [23] S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math., 34 (1981), 481-524. Google Scholar [24] D. Lannes, Sharp estimates for pseudo-differential operators with symbols of limited smoothness and commutators, J. Funct. Anal., 232 (2006), 495-539. Google Scholar [25] D. Lannes, Water Waves: Mathematical Theory and Asymptotics, Mathematical Surveys and Monographs, vol 188 (2013), AMS, Providence.Google Scholar [26] F. Linares and C. Matheus, Well-posedness for the 1-D Zakharov system, Advances in Diff. Equations, 14 (2009), 261-288. Google Scholar [27] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod; Gauthier-Villars, Paris, 1989.Google Scholar [28] A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences volume 53, Springer-Verlag, New-York, 1984.Google Scholar [29] T. Mizumachi, Stability of line solitons for the KP-Ⅱ equation in ${{\mathbb{R}}^{2}}$, Memoirs of the AMS, vol. 238, number 1125, (2015).Google Scholar [30] T. Mizumachi and N. Tzvetkov, Stability of the line soliton of the KP-Ⅱ equation under periodic transverse perturbations, Math. Ann., 352 (2012), 659-690. Google Scholar [31] C. Obrecht, Thèse de Doctorat, Université Paris-Sud (2015) and article in preparation.Google Scholar [32] M. Ohta, Stability and instability of standing waves for the generalized Davey-Stewartson system, Diff. Int. Eq., 8 (1995), 1775-1788. Google Scholar [33] M. Ohta, Instability of standing waves for the generalized Davey-Stewartson system, Ann. Inst. Poincaré H., Phys. Théor., 62 (1995), 69-80. Google Scholar [34] M. Ohta, Blow-up solutions and strong instability of standing waves for the generalized DaveyStewartson system, Ann. Inst. Poincaré H., Phys. Théor., 63 (1995), 111-117. Google Scholar [35] F. Oliveira, Stability of the solitons for the one-dimensional Zakharov-Rubenchik system, Physica D, 175 (2003), 220-240. Google Scholar [36] F. Oliveira, Adiabatic limit of the Zakharov-Rubenchik system, Reports on Mathematical Physics, 61 (2008), 13-27. Google Scholar [37] T. Ozawa and Y. Tsutsumi, Existence and smoothing effect of solutions for the Zakharov equations, Publ. Res. Inst. Math. Sci., Kyoto University. Research Institute for Mathematical Sciences. Publications, 28 (1992), 329-361. Google Scholar [38] T. Ozawa and Y. Tsutsumi, The nonlinear Schrödinger limit and the initial layer of the Zakharov equations, Proc. Jap. Acad. A, 67 (1991), 113-116. Google Scholar [39] T. Ozawa and Y. Tsutsumi, The nonlinear Schrödinger limit and the initial layer of the Zakharov equations, Diff. Int. Eq., 5 (1992), 721-745. Google Scholar [40] T. Passot, P.-L. Sulem and C. Sulem, Generalization of acoustic fronts by focusing ave packets, Physica D, 94 (1996), 168-187. Google Scholar [41] G. Ponce and J.-C. Saut, Well-posedness for the Benney-Roskes-Zakharov-Rubenchik system, Discrete Cont. Dynamical Systems, 13 (2005), 811-825. Google Scholar [42] F. Rousset and N. Tzvetkov, Transverse instability of the line solitary water-waves, Invent. Math., 184 (2011), 257-388. Google Scholar [43] F. Rousset and N. Tzvetkov, Transverse nonlinear instability for two-dimensional dispersive models, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 477-496. Google Scholar [44] F. Rousset and N. Tzvetkov, Transverse nonlinear instability of solitary waves for some Hamiltonian PDE's, J. Math. Pures et Appl., 80 (2008), 550-590. Google Scholar [45] S. H. Schochet and M.I. Weinstein, The nonlinear Schrödinger limit of the Zakharov governing Langmuir turbulence, Comm. Math. Phys., 106 (1986), 569-580. Google Scholar [46] C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation, Springer-Verlag, Applied Mathematical Sciences 139 New York, Berlin, 1999.Google Scholar [47] C. Sulem and P.-L. Sulem, Quelques résultats de régularité pour les équations de la turbulence de Langmuir, C.R. Ac. Sci. Paris Sér. A-B, 289 (1979), A173-A176. Google Scholar [48] H. Takaoka, Well-posedness for the Zakharov system with the periodic boundary condition, Diff. and Int. equations, 6 (1999), 789-810. Google Scholar [49] N. Tzvetkov, Low regularity solutions for a generalized Zakharov system, Diff. and Int. Equations, 13 (2000), 423-440. Google Scholar [50] V. E. Zakharov, Collapse of Langmuir waves, Sov. Phys. JETP, 35 (1972), 908-914. Google Scholar [51] V. E. Zakharov, Weakly nonlinear waves on the surface of an ideal finite depth fluid, Amer. Math. Soc. Transl., 182 (1998), 167-197. Google Scholar [52] V. E. Zakharov and E. A. Kuznetsov, Hamiltonian formalism for nonlinear waves, PhysicsUspekhi, 40 (1997), 1087-1116. Google Scholar [53] V. E. Zakharov and A. M. Rubenchik, Nonlinear interaction of high-frequency and low frequency waves, Prikl. Mat. Techn. Phys., 5 (1972), 84-98. Google Scholar [54] Xiaofei Zhao and Ziyi Li, Numerical methods and simulations for the dynamics of onedimensional Zakharov-Rubenchik equations, J. Sci. Comput., 59 (2014), 412-438. Google Scholar
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