Xiaoxiao ZHENG
,
Yadong SHANG
,
Xiaoming PENG
. ORBITAL STABILITY OF PERIODIC TRAVELING WAVE SOLUTIONS TO THE GENERALIZED ZAKHAROV EQUATIONS[J]. Acta mathematica scientia, Series B, 2017
, 37(4)
: 998
-1018
.
DOI: 10.1016/S0252-9602(17)30054-1
[1] Hadouaj H, Malomed B A, Maugin G A. Dynamics of a soliton in a generalized Zakharov system with dissipation. Phys Rev A, 1991, 44(6):3925-3931
[2] Malomed B, Anderson D, Lisak M, Quiroga-Teixeiro M L. Dynamics of solitary waves in the Zakharov model equations. Physical Rev E, 1997, 55(1):962-968
[3] Zhang J. Variational approach to solitary wave solution of the generalized Zakharov equation. Comput Math Appl, 2007, 54(7/8):1043-1046
[4] Javidi M, Golbabai A. Construction of a solitary wave solution for the generalized Zakharov equation by a variational iteration method. Comput Math Appl, 2007, 54(7/8):1003-1009
[5] Khan Y, Faraz N, Yildirim A. New soliton solutions of the generalized Zakharov equations using He's variational approach. Appl Math Lett, 2011, 24(6):965-968
[6] Betchewe G, Thomas B, Victor K K, Crepin K T. Dynamical survey of a generalized-Zakharov equation and its exact travelling wave solutions. Appl Math Comput, 2010, 217(1):203-211
[7] Wang M L, Li X Z. Extended F-expansion method and periodic wave solutions for the generalized Zakharov equations. Phys Lett A, 2005, 343(1/3):48-54
[8] El-Wakil S A, Degheidy A R, Abulwafa E M, Madkour M A, Attia M T, Abdou M A. Exact travelling wave solutions of generalized Zakharov equations with arbitrary power nonlinearities. Int J Nonlinear Sci, 2009, 7(4):455-461
[9] Suarez P, Biswas A. Exact 1-soliton solution of the Zakharov equation in plasmas with power law nonlinearity. Appl Math Comput, 2011, 217(17):7372-7375
[10] Zedan H A. G'/G-expansion method for the generalized Zakharov equations. Ric Mat, 2011, 60(2):203-217
[11] Song M, Liu Z R. Traveling wave solutions for the generalized Zakharov equations. Math Probl Eng, 2012, Art ID 747295
[12] Fang S M, Guo C H, Guo B L. Exact traveling wave solutions of modified Zakharov equations for plasmas with a quantum correction. Acta Math Sci, 2012, 32B(3):1073-1082
[13] You S J, Guo B L, Ning X Q. Initial boundary value problem for modified Zakharov equations. Acta Math Sci, 2012, 32B(4):1455-1466
[14] You S J. The posedness of the periodic initial value problem for generalized Zakharov equations. Nonlinear Anal, 2009, 71(7/8):3571-3584
[15] Yang H. Orbital stability of solitary waves for the generalized Zakharov system. J Partial Diff Eqs, 2007, 20:252-264
[16] Yang H. Orbital stability of solitary waves for the generalized Zakharov system. Adv Math (China), 2006, 35(5):635-637
[17] Ohta M. Stability of solitary waves for the Zakharov equations Dynamical Systems and Applications. World Scientific Series in Applied Analysis Vol 4. River Edge, NJ:World Scientific, 1995:563-571
[18] Wu Y P. Orbital stability of solitary waves of Zakharov system. J Math Phys, 1994, 35:2413-2422
[19] Angulo J, Banquet C. Orbital stability for the periodic Zakharov system. Nonlinearity, 2011, 24:2913-2932
[20] Benjamin T. The stability of solitary waves. Proc R Soc Lond A, 1972, 338:153-183
[21] Bona J. On the stability theory of solitary waves. Proc R Soc Lond Ser A, 1975, 344:363-374
[22] Weinstein M. Modulation stability of ground states of nonlinear Schrödinger equation. SIAM J Math Anal, 1985, 16:472-490
[23] Weinstein M. Lyapunov stability of ground states of nonlinear dispersive equations. Commun Pure Appl Math, 1986, 39:51-68
[24] Iorio R Jr, Iorio V. Fourier analysis and partial differential equations. Cambridge Studies in Advanced Mathematics, 2001, vol 70, Cambridge:Cambridge University Press
[25] Byrd P, Friedman M. Handbook of Elliptic Integrals for Engineers and Scientists. 2nd ed. New York:Springer, 1971
[26] Wang Z X, Guo D R. Special Functions. Singapore:World Scientific Publishing, 1989
[27] Ince E L. The periodic Lamé functions. Proc Roy Soc Edinburgh, 1940, 60:47-63
[28] Magnus W, Winkler S. Hill's Equation. Tracts Pure Appl Math, vol 20, New York:Wiley, 1976
[29] Weinstein M. Modulation stability of ground states of nonlinear Schrödinger equation. SIAM J Math Anal, 1985, 16:472-490
[30] Angulo J. Nonlinear Dispersive Evolution Equations:Existence and Stability of Solitary and Periodic Traveling Waves Solutions (Mathematical Surveys and Monographs Series (SURV vol 156)). Providence, RI:American Mathematical Society, 2009
First, we prove the existence of a smooth curve of positive traveling wave solutions of dnoidal type with a fixed fundamental period L for the generalized Zakharov equations. Then, by using the classical method proposed by Benjamin, Bona et al., we show that this solution is orbitally stable by perturbations with period L. The results on the orbital stability of periodic traveling wave solutions for the generalized Zakharov equations in this paper can be regarded as a perfect extension of the results of[15, 16, 19].