Hamiltonian Conservation Laws and Finite-Difference Simulations for a Nonlinearly Coupled Generalized Schrödinger–KdV System

Hongyi Wu

doi:10.54097/rrw7wr47

Authors

Hongyi Wu Department of Mathematics, University of California, Berkeley, Berkeley, CA, USA

DOI:

https://doi.org/10.54097/rrw7wr47

Keywords:

Schrödinger–KdV system; long-short wave interaction; Hamiltonian structure; conservation law; finite difference method; implicit midpoint rule.

Abstract

This paper studies a Hamiltonian generalized Schrödinger–KdV system generated by a nonlinear long-wave coupling potential. Starting from an energy functional with a smooth coupling function , we derive a matched coupled system involving the terms and , with the quadratic case used for numerical simulations. Under periodic boundary conditions, we prove conservation of the Schrödinger mass, the long-wave mean, a generalized momentum, and the Hamiltonian energy. A fully discrete finite-difference method based on central differences and the implicit midpoint rule is then constructed. The scheme exactly preserves the discrete mass and discrete long-wave mean, while the generalized momentum and energy are monitored as diagnostic quantities. Numerical experiments confirm the expected second-order convergence behavior and illustrate the long-time boundedness of the monitored invariant errors.

References

[1] K. Nishikawa, H. Hojo, K. Mima, and H. Ikezi, Coupled nonlinear electron-plasma and ion-acoustic waves, Physical Review Letters, 33 (1974), 148–151.

[2] T. Kawahara, N. Sugimoto, and T. Kakutani, Nonlinear interaction between short and long capillary-gravity waves, Journal of the Physical Society of Japan, 39 (1975), 1379–1386.

[3] D. J. Benney, A general theory for interactions between short and long waves, Studies in Applied Mathematics, 56 (1977), 81–94.

[4] K. Appert and J. Vaclavik, Dynamics of coupled solitons, Physics of Fluids, 20 (1977), 1845–1849.

[5] A. Arbieto, A. J. Corcho, and C. Matheus, Rough solutions for the periodic Schrödinger–Korteweg–de Vries system, Journal of Differential Equations, 230 (2006), 295–336.

[6] A. J. Corcho and F. Linares, Well-posedness for the Schrödinger–Korteweg–de Vries system, Transactions of the American Mathematical Society, 359 (2007), 4089–4106.

[7] C. Matheus, Global well-posedness of NLS-KdV systems for periodic functions, Electronic Journal of Differential Equations, 2007, no. 7, 1–20.

[8] Z. Guo and Y. Wang, On the well-posedness of the Schrödinger–Korteweg–de Vries system, Journal of Differential Equations, 249 (2010), 2500–2520.

[9] F. S. de Oliveira, Well-posedness and existence of bound states for a coupled Schrödinger-gKdV system, Nonlinear Analysis: Theory, Methods & Applications, 73 (2010), 2686–2698.

[10] J. P. Albert and S. Bhattarai, Existence and stability of a two-parameter family of solitary waves for an NLS-KdV system, arXiv preprint arXiv:1212.3902, 2012.

[11] S. Bhattarai, A. J. Corcho, and M. Panthee, Well-posedness for multicomponent Schrödinger-gKdV systems and stability of solitary waves with prescribed mass, arXiv preprint arXiv:1509.08188, 2015.

[12] D. Bai and L. Zhang, The finite element method for the coupled Schrödinger-KdV equations, Physics Letters A, 373 (2009), 2237–2244.

[13] A. Golbabai and A. Safdari-Vaighani, A meshless method for numerical solution of the coupled Schrödinger-KdV equations, Computing, 92 (2011), 225–242.

[14] P. Amorim and M. Figueira, Convergence of a numerical scheme for a coupled Schrödinger–KdV system, Revista Matemática Complutense, 26 (2013), 409–426.

[15] S. Xu and J. Yan, A conservative local discontinuous Galerkin method for the Schrödinger-KdV system, Communications in Computational Physics, 14 (2013), 1287–1310.

[16] J. Qiu, D. Han, and H. Zhou, A general conservative eighth-order compact finite difference scheme for the coupled Schrödinger-KdV equations, AIMS Mathematics, 8 (2023), 10596–10618.

[17] O. Gonzalez, Time integration and discrete Hamiltonian systems, Journal of Nonlinear Science, 6 (1996), 449–467.

[18] R. I. McLachlan, G. R. W. Quispel, and N. Robidoux, Geometric integration using discrete gradients, Philosophical Transactions of the Royal Society A, 357 (1999), 1021–1045.

[19] G. R. W. Quispel and D. I. McLaren, A new class of energy-preserving numerical integration methods, Journal of Physics A: Mathematical and Theoretical, 41 (2008), 045206.

[20] E. Hairer, C. Lubich, and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd ed., Springer, Berlin, 2006.

[21] E. Celledoni, V. Grimm, R. I. McLachlan, D. I. McLaren, D. O’Neale, B. Owren, and G. R. W. Quispel, preserving energy resp. dissipation in numerical PDEs using the average vector field method, Journal of Computational Physics, 231 (2012), 6770–6789.