Conservative compact difference scheme for the Zakharov–Rubenchik equations

References

• W. Bao, F. Sun, and G. Wei, Numerical methods for the generalized Zakharov system, J. Comput. Phys. 190 (2003), pp. 201–228. doi: 10.1016/S0021-9991(03)00271-7
   Web of Science ®Google Scholar
• A. 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
• Q.S. Chang, B.L. Guo, and H. Jiang, Finite difference method for generalized Zakharov equations, Math. Comput. 64 (1995), pp. 537–553. doi: 10.1090/S0025-5718-1995-1284664-5
   Web of Science ®Google Scholar
• J. Colliander, Wellposedness for Zakharov systems with generalized nonlinearity, J. Differ. Equ. 148 (1998), pp. 351–363. doi: 10.1006/jdeq.1998.3445
   Web of Science ®Google Scholar
• W. Dai, A new compact finite difference scheme for solving the complex Ginzburg-Landau equation, Appl. Math. Comput. 260 (2015), pp. 268–287.
   Web of Science ®Google Scholar
• Y. Gong, Y. Long, U. Wang, and X. Deng, LU method for solving block tridiagonal matrix equations and its applications, J. Dalian Univ. Tech. 4 (1997), pp. 406–409. (in chinese).
   Google Scholar
• Y. Gong, J. Cai, and Y. Wang, Some new structure-preserving algorithms for general multi-symplectic formulations of Hamiltonian PDEs, J. Comput. Phys. 279 (2014), pp. 80–102. doi: 10.1016/j.jcp.2014.09.001
   Web of Science ®Google Scholar
• B. Guo, P. Paseual, M. Rodriguez, and L. Vázquez, Numerical solution of the Sine-Gorden equation, Appl. Math. Comput. 18 (1986), pp. 1–14.
   Web of Science ®Google Scholar
• S. Li and L. Vu-Quoc, Finite difference calculus invariant structure of a class of algorithms for the nonlinear Klein-Gordon equation, SIAM J. Numer. Anal 32 (1995), pp. 1839–1875. doi: 10.1137/0732083
   Web of Science ®Google Scholar
• F. Liao and L. Zhang, Conservative compact finite difference scheme for the Coupled Scherödinger-Boussinesq Equation, Numer. Methods Partial Differ. Equ. 32 (2016), pp. 1667–1688. doi: 10.1002/num.22067
   Web of Science ®Google Scholar
• F. Linares and C. Matheus, Well-posedness for the 1D Zakharov-Rubenchik system, Adv. Differ. Equ. 14 (2009), pp. 261–288.
   Web of Science ®Google Scholar
• F. Oliveira, Stability of the solitons for the one-dimensional Zakharov-Rubenchik equation, Phys. D 175 (2003), pp. 220–240. doi: 10.1016/S0167-2789(02)00722-4
   Web of Science ®Google Scholar
• F. Oliveira, Adiabatic limit of the Zakharov-Rubenchik equation, Rep. Math. Phys. 61 (2008), pp. 13–27. doi: 10.1016/S0034-4877(08)00006-2
   Web of Science ®Google Scholar
• G. Ponce and J. Saut, Well-posedness for the Benney-Zakharov-Rubenchik system., Discret. Contin. Dyn. Syst. 13 (2005), pp. 818–852.
   Web of Science ®Google Scholar
• T. Wang, Optimal point-wise error estimate of a compact difference scheme for the coupled nonlinear Schrödinger equations, J. Comput. Math. 32(1) (2014), pp. 58–74. doi: 10.4208/jcm.1310-m4340
   Web of Science ®Google Scholar
• T. Wang, Optimal point-wise error estimate of a compact difference scheme for the coupled Gross-Pitaevskii equations in one dimension, J. Sci. Comput. 59 (2014), pp. 158–186. doi: 10.1007/s10915-013-9757-1
   Web of Science ®Google Scholar
• T. Wang, Uniform pointwise error estimates of semi-implicit compact finite difference methods for the nonlinear Schrödinger equation perturbed by wave operator, J. Math. Anal. Appl. 422 (2015), pp. 286–308. doi: 10.1016/j.jmaa.2014.08.026
   Web of Science ®Google Scholar
• T. Wang and X. Zhao, Optimal l∞ error estimates of finite difference methods for the coupled Gross–Pitaevskii equations in high dimensions, Sci. China Math. 57 (2014), pp. 2189–2214. doi: 10.1007/s11425-014-4773-7
   Web of Science ®Google Scholar
• T. Wang, B. Guo, and L. Zhang, New conservative difference schemes for a coupled nonlinear Schrödinger system, Appl. Math. Comput. 217 (2010), pp. 1604–1619.
   Web of Science ®Google Scholar
• V. Zakharov and A. Rubenchik, Nonlinear interaction between high and low frequency waves, Prikl. Mat. Techn. Fiz. 5 (1972), pp. 84–89.
   Google Scholar
• F. Zhang, Numerical simulation of nonlinear Schrödinger equation system: A new conservative scheme, Appl. Math. Comput. 71 (1995), pp. 165–177.
   Web of Science ®Google Scholar
• X. Zhao and Z. Li, Numerical methods and simulations for the dynamics of one-dimensional Zakharov–Rubenchik equations, J. Sci. Comput. 59 (2014), pp. 412–438. doi: 10.1007/s10915-013-9768-y
   Web of Science ®Google Scholar
• Y. Zhou, Application of Discrete Functional Analysis to the Finite Difference Methods, International Academic Publishers, Beijing, 1990.
   Google Scholar