Journal of Applied Mathematics

Existence and Uniqueness of Solutions for a Type of Generalized Zakharov System

Rui Li, Xing Lin, Zongwei Ma, and Jingjun Zhang

Full-text: Open access


We study the Cauchy problem for a type of generalized Zakharov system. With the help of energy conservation and approximate argument, we obtain global existence and uniqueness in Sobolev spaces for this system. Particularly, this result implies the existence of classical solution for this generalized Zakharov system.

Article information

J. Appl. Math., Volume 2013 (2013), Article ID 193589, 7 pages.

First available in Project Euclid: 14 March 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Li, Rui; Lin, Xing; Ma, Zongwei; Zhang, Jingjun. Existence and Uniqueness of Solutions for a Type of Generalized Zakharov System. J. Appl. Math. 2013 (2013), Article ID 193589, 7 pages. doi:10.1155/2013/193589.

Export citation


  • V. E. Zakharov, “Collapse of Langmuir waves,” Journal of Exper-imental and Theoretical Physics, vol. 35, pp. 908–914, 1972.
  • H. Added and S. Added, “Existence globale de solutions fortes pour les equations de la turbulence de Langmuir en dimension 2,” Comptes Rendus de l'Académie des Sciences, vol. 299, pp. 551–554, 1984.
  • B. L. Guo and L. J. Shen, “The existence and uniqueness of the classical solution to the periodic initial value problem for the Zakharov equation,” Acta Mathematicae Applicatae Sinica, vol. 5, no. 3, pp. 310–324, 1982 (Chinese).
  • T. Ozawa and Y. Tsutsumi, “Existence and smoothing effect ofsolutions for the Zakharov equations,” Kyoto University. Rese-arch Institute for Mathematical Sciences. Publications, vol. 28, no. 3, pp. 329–361, 1992.
  • C. Sulem and P.-L. Sulem, “Quelques résultats de régularité pour les équations de la turbulence de Langmuir,” Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences A et B, vol. 289, no. 3, pp. A173–A176, 1979.
  • S. H. Schochet and M. I. Weinstein, “The nonlinear Schrödinger limit of the Zakharov equations governing Langmuir turbulence,” Communications in Mathematical Physics, vol. 106, no. 4, pp. 569–580, 1986.
  • J. Ginibre, Y. Tsutsumi, and G. Velo, “On the Cauchy problem for the Zakharov system,” Journal of Functional Analysis, vol. 151, no. 2, pp. 384–436, 1997.
  • I. D. Chueshov and A. S. Shcherbina, “On 2D Zakharov system in a bounded domain,” Differential and Integral Equations, vol. 18, no. 7, pp. 781–812, 2005.
  • I. Flahaut, “Attractors for the dissipative Zakharov system,” Non-linear Analysis: Theory, Methods & Applications, vol. 16, no. 7-8, pp. 599–633, 1991.
  • O. Goubet and I. Moise, “Attractor for dissipative Zakharov sys-tem,” Nonlinear Analysis: Theory, Methods & Applications, vol. 31, no. 7, pp. 823–847, 1998.
  • Y. Li and B. Guo, “Attractor of dissipative radially symmetric Zakharov equations outside a ball,” Mathematical Methods in the Applied Sciences, vol. 27, no. 7, pp. 803–818, 2004.
  • B. Guo, J. Zhang, and X. Pu, “On the existence and uniqueness of smooth solution for a generalized Zakharov equation,” Journal of Mathematical Analysis and Applications, vol. 365, no. 1, pp. 238–253, 2010.
  • B. Guo and J. Zhang, “Well-posedness of the Cauchy problem for the magnetic Zakharov type system,” Nonlinearity, vol. 24, no. 8, pp. 2191–2210, 2011.
  • L. Han, J. Zhang, Z. Gan, and B. Guo, “Cauchy problem for the Zakharov system arising from hot plasma with low regularity data,” Communications in Mathematical Sciences, vol. 11, no. 2, pp. 403–420, 2013.
  • J. Zhang, C. Guo, and B. Guo, “On the Cauchy problem for the magnetic Zakharov system,” Monatshefte für Mathematik, vol. 170, no. 1, pp. 89–111, 2013.
  • N. Laskin, “Fractional quantum mechanics and Lévy path integ-rals,” Physics Letters A, vol. 268, no. 4–6, pp. 298–305, 2000.
  • N. Laskin, “Fractional quantum mechanics,” Physical Review E, vol. 62, no. 3, pp. 3135–3145, 2000.
  • B. Guo and Z. Huo, “Global well-posedness for the fractional nonlinear Schrödinger equation,” Communications in Partial Differential Equations, vol. 36, no. 2, pp. 247–255, 2011.
  • B. Guo, Y. Han, and J. Xin, “Existence of the global smooth solu-tion to the period boundary value problem of fractional non-linear Schrödinger equation,” Applied Mathematics and Computation, vol. 204, no. 1, pp. 468–477, 2008.
  • R. Coifman and Y. Meyer, “Nonlinear harmonic analysis operator theory and P.D.E.,” in Beijing Lectures in Harmonic Analysis, pp. 3–45, Princeton University Press, 1986.
  • T. Kato, Liapunov Functions and Monotonicity in the Euler andNavier-Stokes Equations, vol. 1450 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1990.
  • C. E. Kenig, G. Ponce, and L. Vega, “Well-posedness of the initi-al value problem for the Korteweg-de Vries equation,” Journal of the American Mathematical Society, vol. 4, no. 2, pp. 323–347, 1991.