Conservative difference scheme for fractional Zakharov system and convergence analysis

References

• S. Abbasbandy, E. Babolian, M. Ashtiani, Numerical solution of the generalized Zakharov equation by homotopy analysis method. Commun. Nonlinear Sci. Numer. Simul 14 (2009), pp. 4114–4121.
   Web of Science ®Google Scholar
• E.A-B. Abdel-Salam and Z.I.A. Al-Muhiameed, Analytical solutions of the space-time fractional quantum Zakharov system for plasmas, Am. J. Comput. Appl. Math 6 (2016), pp. 19–30.
   Google Scholar
• W. Bao and F. Sun, Efficient and stable numerical methods for the generalized and vector Zakharov system, SIAM J. Sci. Comput. 26 (2005), pp. 1057–1088.
   Web of Science ®Google Scholar
• A.H. Bhrawy, An efficient Jacobi pseudospectral approximation for nonlinear complex generalized Zakharov system, Appl. Math. Comput. 247 (2014), pp. 30–46.
   Web of Science ®Google Scholar
• J. Colliander, Wellposedness for Zakharov systems with generalized nonlinearity, J. Differ. Equ. 148 (1988), pp. 351–363.
   Google Scholar
• A.S. Davydov, Solitons in molecular systems, Phys. Scr. 20 (1979), pp. 387–394.
   Web of Science ®Google Scholar
• L.M. Degtyarev, V.G. Nakhan-kov, and L.I. Rudakov, Dynamics of the formation and interaction of Langmuir solitons and strong turbulence, Sov. Phys.-JETP 40 (1974), pp. 264–268.
   Google Scholar
• M. Dehghan and M. Abbaszadeh, Numerical investigation based on direct meshless local Petrov Galerkin (direct MLPG) method for solving generalized Zakharov system in one and two dimensions and generalized Gross-Pitaevskii equation, Eng. Comput.-Germany 33 (2017), pp. 983–996.
   Web of Science ®Google Scholar
• F. Demengel and G. Demengel, Functional Spaces for the Theory of Elliptic Partial Differential Equations, Springer, London, 2012.
   Google Scholar
• S. Duo and Y. Zhang, Mass-Conservative Fourier spectral methods for solving the fractional nonlinear Schrödinger equation, Comput. Math. Appl. 71 (2016), pp. 2257–2271.
   Web of Science ®Google Scholar
• V.J. Ervin and J.P. Roop, Variational formulation for the stationary fractional advection dispersion equation, Numer. Methods Partial Differ. Equ. 22 (2006), pp. 558–576.
   Web of Science ®Google Scholar
• G. Hardy, J. Littlewood, and G. Polya, Inequalities, Cambridge University Press, London, 1952.
   Google Scholar
• M. Javidi and A. Golbabai, Construction of a solitary wave solution for the generalized Zakharov equation by a variational iteration method, Comput. Math. Appl. 54 (2007), pp. 1003–1009.
   Web of Science ®Google Scholar
• Y. Ji and H. Ma, Uniform convergence of the legendre spectral method for the Zakharov equations, Numer. Methods Partial Differ. Equ. 29 (2013), pp. 475–495.
   Web of Science ®Google Scholar
• S. Jin and C. Zheng, A time-splitting spectral method for the generalized Zakharov system in multi-dimensions, J. Sci. Comput. 26 (2006), pp. 127–149.
   Web of Science ®Google Scholar
• K. Kirkpatrick, E. Lenzmann, and G. Staffilani, On the continuum limit for discrete NLS with long-range lattice interactions, Commun. Math. Phys. 317 (2013), pp. 563–591.
   Web of Science ®Google Scholar
• M. Li, X. Gu, C. Huang, M. Fei, and G. Zhang, A fast linearized conservative finite element method for the strongly coupled nonlinear fractional Schrödinger equations, J. Comput. Phys. 358 (2018), pp. 256–282.
   Web of Science ®Google Scholar
• J.E. Macías-Díaz, A.S. Hendy, and R.H. De Staelen, A compact fourth-order in space energy-preserving method for Riesz space-fractional nonlinear wave equations, Appl. Math. Comput. 325 (2018), pp. 1–14.
   Web of Science ®Google Scholar
• K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.
   Google Scholar
• M.D. Ortigueira, Riesz potential operators and inverses via fractional centred derivatives. Int. J. Math. Math. Sci. 2006 (2006), pp. 1–12.
   Google Scholar
• X. Pan and L. Zhang, On the convergence of a high-accuracy conservative scheme for the Zakharov equations, Appl. Math. Comput. 297 (2017), pp. 79–91.
   Web of Science ®Google Scholar
• G.L. Payne, D.R. Nicholson, and R.M. Downie, Numerical solution of the Zakharov equations, J. Comput. Phys. 50 (1983), pp. 482–498.
   Web of Science ®Google Scholar
• N.R. Pereira, Collisions between Langmuir solitons, Phys. Fluids 20 (1977), pp. 750–755.
   Web of Science ®Google Scholar
• I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, New York, 1999.
   Google Scholar
• L. Stenflo, Nonlinear equations for acoustic gravity waves, Phys. Scr. 33 (1986), pp. 156–158.
   Web of Science ®Google Scholar
• Z. Sun, Numerical Methods of the Partial Differential Equation. 2nd ed. Science Press, Beijing, 2012. (in Chinese).
   Google Scholar
• J. Wang, Multisymplectic numerical method for the Zakharov system, Comput. Phys. Commun. 180 (2009), pp. 1063–1071.
   Web of Science ®Google Scholar
• P. Wang and C. Huang, An energy conservative difference scheme for the nonlinear fractional Schrödinger equation, J. Comput. Phys. 293 (2015), pp. 238–251.
   Web of Science ®Google Scholar
• X. Wang and Y. Shang, On the global existence and small dissipation limit for generalized dissipative Zakharov system, Math. Methods Appl. Sci. 41 (2018), pp. 3718–3749.
   Web of Science ®Google Scholar
• J. Wang and A. Xiao, An efficient conservative difference scheme for fractional Klein-Gordon-Schrödinger equations, Appl. Math. Comput. 320 (2018), pp. 691–709.
   Web of Science ®Google Scholar
• D. Wang, A. Xiao, and W. Yang, Maximum-norm error analysis of a difference scheme for the space fractional CNLS, Appl. Math. Comput. 257 (2015), pp. 241–251.
   Web of Science ®Google Scholar
• Y. Xia, Y. Xu, and C.W. Shu, Local discontinuous Galerkin methods for the generalized Zakharov system, J. Comput. Phys. 229 (2010), pp. 1238–1259.
   Web of Science ®Google Scholar
• Q. Yang, F. Liu, and I. Turner, Numerical methods for fractional partial differential equations with Riesz space fractional derivatives, Appl. Math. Model 34 (2010), pp. 200–218.
   Web of Science ®Google Scholar
• S. You and X. Ning, Well-posedness for a class of generalized Zakharov system, J. Nonlinear Sci. Appl.10 (2017), pp. 1289–1302.
   Google Scholar
• V.E. Zakharov, Collapse of Langmuir waves, Sov. Phys.-JETP 35 (1972), pp. 908–914.
   Google Scholar
• J. Zhang, Variational approach to solitary wave solution of the generalized Zakharov equation, Comput. Math. Appl. 54 (2007), pp. 1043–1046.
   Web of Science ®Google Scholar
• X. Zhao, Z. Sun, and Z. Hao, A fourth-order compact ADI scheme for two-dimensional nonlinear space fractional Schrödinger equation, SIAM J. Sci. Comput. 36 (2014), pp. A2865–A2886.
   Web of Science ®Google Scholar
• X. Zhou and L. Zhang, A conservative compact difference scheme for the Zakharov equations in one space dimension, Int. J. Comput. Math. 95 (2018), pp. 279–302.
   Web of Science ®Google Scholar