New Conservative Schemes for Zakharov Equation

Year 2023, Volume: 15 Issue: 2, 277 - 293, 31.12.2023

Ayhan Aydın , Bahaa Ahmed Khalaf Sabawe

https://doi.org/10.47000/tjmcs.1226770

Abstract

New first-order and second-order energy preserving schemes are proposed for the Zakharov system. The methods are fully implicit and semi-explicit. It has been found that the first order method is also massconserving. Concrete schemes have been applied to simulate the soliton evolution of the Zakharov system. Numerical results show that the proposed methods capture the remarkable features of the Zakharov equation. We have obtained that the semi-explicit methods are more efficient than the fully implicit methods. Numerical results also demonstrate that the new energy-preserving schemes accurately simulate the soliton evolution of the Zakharov system.

Keywords

Average vector field method, partitioned average vector field method, Hamiltonian system, Zakharov equation

References

• Aderogba, A.A., Appadu, A.R., Classical and multisymplectic schemes for linearized KdV equation: Numerical results and dispersion analysis, Fluids, 6(2021), 214.
• Akkoyunlu, C., Karasözen, B., , Average Vector Field Splitting Method for Nonlinear Schr¨odinger Equation, Chaos and Complex Systems, (2013), 245–251.
• Aktürk, T. , Bulut, H. , Yel, G., An application of the modified expansion method to nonlinear partial differential equation, Turkish Journal of Mathematics and Computer Science, 10(2018), 202–206.
• Anderson, D., Variational approach to nonlinear pulse propagation in optical fibers, Phys. Rev. A, 27(6)(1983), 3135–3145.
• Appadu, A.R., Kelil, A.S., On semi-analytical solutions for linearized dispersive KdV equations, Mathematics, 8(2020), 1769.
• Appadu, A.R., Kelil, A.S., Comparison of modified ADM and classical finite difference method for some third-order and fifth-order KdV equations, Demonstratio Mathematica, 54(2021), 377–409.
• Aydin, A., Multisymplectic integration of N-coupled nonlinear Schr¨odinger equation with destabilized periodic wave solutions, Chaos, Solitons and Fractals, 41(2)(2009), 735–751
• Aydin, A., A convergent two-level linear scheme for the generalized Rosenau-KdV-RLW equation Turkish Journal of Mathematics, 43(5)(2019), 2226–2245.
• Aydin, A., Koroglu, C., A nonstandard numerical method for the modified KdV equation Pramana 89(5)(2017), 1–6.
• Aydin, A., Karasözen, B., Lobatto IIIA-IIIB discretization of the strongly coupled nonlinear Schr¨odinger equation, Journal of Computational and Applied Mathematics, 235(2011), 4770–4779.
• Aydin, A., Karasözen, B., Symplectic and multi-symplectic methods for coupled nonlinear Schr¨odinger equations with periodic solutions, Computer Physics Communications, 177(2007), 566–583.
• Aydin, A., Karasözen, B., Multi-symplectic integration of coupled nonlinear Schr¨odinger system with soliton solutions, International Journal of Computer Mathematics, 86(2009), 864–882.
• Bailung, H., Sharma, S.K., Nakamura, Y., Observation of Peregrine solitons in a multicomponent plasma with negative ions, Phys. Rev. Lett., 107(2011), 255005.
• Bao, W., Sun, F.F., Wei, G.V., Numerical methods for the generalized Zakharov system, J. Comput. Phys., 190(2003), 201–228.
• Baohui, H., Dong,L., Energy-preserving time high-order AVF compact finite difference schemes for nonlinear wave equations with variable coefficients, J. Comp. Phys., 421(2020), 109738.
• Cai, W., Li, H., Wang, Y., Partitioned average vector field methods, Journal of Computational Physics, 37(2018), 25–42.
• Celledoni, E., Grimn, V., McLachlan, R.I., McLaren, D.I., O’Neale, D. et al., Preserving energy rep. dissipation in numerical PDEs using average vector field method, Journal of Computational Physics, 231(2012), 6770–6789.
• Chang, Q., Jiang, H., A conservative difference scheme for the Zakharov equations, J. Comput. Phys., 113(1994), 309–319.
• Degtyarev, L.M., Nakhankov, V.G., Rudakov, L.I., Dynamics of the formation and interaction of Langmuir solitons and strong turbulence, Sov Phys JETP, 40(2)(1974), 532–542.
• Demiray, S.T., Bulut, H., New exact solutions of the system of equations for the ion sound and Langmuir waves by ETEM, Math. Comput. Appl., 21(2)(2016), 1–9.
• Ganshin, A.N., Efimov, V.B., Kolmakov, G.V., Mezhov-Deglin, L.P., McClintock, P.E.V., Observation of an inverse energy cascade in developed acoustic turbulence in superfluid helium, Phys. Rev. Lett., 101(2008), 065303.
• Glassey, R., Approximate solutions to the Zakharov equations via finite differences, J. Comput. Phys., 100(1992), 377–383.
• Glassey, R., Convergence of an energy-preserving scheme for the Zakharov equations in one space dimension, Math. Comput., 58(1992), 83–102.
• Goldman, M.V., Strong turbulence of plasma waves, Rev. Modern Phys., 56(4)(1984), 709–735.
• Hairer, E., Energy-preserving variant of collocation methods, ESCM, 5(1)(2010), 73–84.
• Hohmann, R., Kuhl, U., Stockmann, H.J., Kaplan, L., Heller E.J., Freak waves in the linear regime: A microwave study, Phys. Rev. Lett., 104(2010), 093901.
• Hong, Q., Wang, J.L., Wang, Y.S, A local energy-preserving scheme for Zakharov system, Chinese Physics B, 27(2)(2018), 020202.
• Iskandarova, G., Kaya, D., Symmetry solution on fractional equation, An Int. J. of Optimization and Control : Theories & Applications, 7(3)(2017), 255–259.
• Jan, L.C., Improving the accuracy of the AVF method, J. Comput. Math., 259(2014), 233–243.
• Ji, B., Zhang, L., Zhou, X., Conservative compact difference scheme for the Zakharov-Rubenchik equations, International Journal of Computer Mathematics, 96(3)(2019), 537–556.
• Kit, E., Shemer, L., Spatial versions of the Zakharov and Dysthe evolution equations for deepwater gravity waves, J. Fluid Mech., 450(2002), 201–205.
• Kocak, H., Kink and anti-kink wave solutios for the generatized KdV equation with Fisher-type nonlinearity, An Int. J. of Optimization and Control : Theories & Applications, 11(2)(2021), 123–127.
• Koroglu, C., Aydin, A., An unconventional finite difference scheme for modified Korteweg-de Vries equation, Advances in Mathematical Physics, 4796070(2017).
• Koroglu, C., Aydin, A., Exact and nonstandard finite difference schemes for the Burgers equation B(2,2), Turk J Math, 45(2021), 647–660.
• Montina, A., Bortolozzo, U., Residori, S., Arecchi, F.T., Non-Gaussian statistics and extreme waves in a nonlinear optical cavity, Phys. Rev. Lett., 103(2009), 173901.
• Ozdemir, N., M-truncated soliton solutions of the fractional (4+1)-dimensional Focal equation, An Int. J. of Optimization and Control : Theories & Applications, 13(1)(2023), 123–129.
• Özdemir, N., Secer, A., Wavelet-based numerical approaches for solving the Korteweg-de Vries (KdV) Equation, Turk. J. Math. Comput. Sci., 14(1)(2022), 44–55.
• Pandir, Y. , Ulusoy, H., Solutions of nonlinear partial differential equations using generalized hyperbolic functions, Turkish Journal of Mathematics and Computer Science, 1(2016), 38–46.
• Payne, G.L., Nicholson, D.R., Downie, R.M., Numerical solution of the Zakharov system, J. Comput. Phys., 50(1983), 482–498.
• Quispel, G.R.W., McLaren, D.I., A new class of energy-preserving numerical integration methods, J. Phys. A: Math. Theo., 41(2008), 045206.
• Solli, D.R., Ropers, C., Koonath, P., Jalali, B., Optical rogue wave, Nature, 450(2007), 1054–1057.
• Tekin, I., Existence and uniqueness of an inverse problem for a second order hyperbolic equation, Universal Journal of Mathematics and Applications, 1(3)(2018), 178–185.
• Tripathy, A., Sahoo, S., Exact solutions for the ion sound Langmuir wave model by using two novel analytical methods, Results in Physics, 19(2020), 103494.
• Uzunca, M., Karas¨ozen, B., Aydin, A., Global energy preserving model reduction for multi-symplectic PDEs, Applied Mathematics and Computation, 436(2023), 127483.
• Wang, J., Multisymplectic numerical method for the Zakharov system, Comp. Phys. Commun., 180(2009), 1063–1071.
• Wang, L., Cai, W., Wang, Y., An energy-preserving scheme for the coupled Gross-Pitaevskii equations, Adv. Appl. Math. Mech., 13(1)(2020), 203–231.
• Yajima, N., Oikawa, M., Formation and interaction of Sonic-Langmuir solitons, Prog Theor Phys, 56(6)(1976), 317–327.
• Zakharov, V.E., Collapse of Langmuir waves, Sov. Phys. JETP, 35(5)(1972), 908–914.