Conservation laws and travelling wave solutions of a generalized NLS-Boussinesq system using symmetries

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Abstract

This work investigates a coupled generalized Schrödinger-Boussinesq system with a p-power nonlinearity. These equations appear in several physical applications. To begin, we present a complete classification of infinitesimal point symmetries for the system. Then, low-order local conservation laws are derived using the multiplier method, providing conserved quantities such as momentum, mass, and charge. By introducing a potential variable, we derive a Lagrangian formulation for the potential system and find nonlocal conservation laws via Noether’s theorem. Finally, we explore travelling wave solutions by reducing the governing equations to a coupled autonomous system of ordinary differential equations. Several families of solutions are constructed, and their physical relevance is discussed.

Keywords

Schrödinger-Boussinesq / symmetries / conservation laws / Noether’s theorem / travelling wave solutions

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References

[1] G. B. Whitham, Linear and Nonlinear Waves, John Wiley & Sons, 1974.

[2] M. J. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform, Society for Industrial and Applied Mathematics, 1981.

[3] S. C. Anco, M. L. Gandarias, E. Recio, Conservation laws, symmetries, and line soliton solutions of generalized KP and Boussinesq equations with p-power nonlinearities in two dimensions, Theor. Math. Phys., 197 (2018), 1393-1411. https://doi.org/10.1134/S004057791810001X

[4] A. P. Márquez, M. L. Gandarias, S. C. Anco, Conservation laws, symmetries, and line solitons of a Kawahara-KP equation, J. Comput. Appl. Math., 436 (2024), 115412. https://doi.org/10.1016/j.cam.2023.115412

[5] K. Nishikawa, H. Hojo, K. Mima, H. Ikezi, Coupled nonlinear electron-plasma and ion-acoustic waves, Phys. Rev. Lett., 33 (1974), 148-151. https://doi.org/10.1103/PhysRevLett.33.148

[6] N. N. Rao, Nonlinear wave modulations in plasmas, Pramana J. Phys., 49 (1997), 109-127. https://doi.org/10.1007/BF02856342

[7] T. Kawahara, N. Sugimoto, T. Kakutani, Nonlinear interaction between short and long capillary-gravity waves, J. Phys. Soc. Jpn., 39 (1975), 1379-1386. https://doi.org/10.1143/JPSJ.39.1379

[8] S. Davydov, Solitons in Molecular Systems, Reidel: Dordrecht, 1987.

[9] A. S. Davydov, A. V. Zolotariuk, Subsonic and supersonic solitons in nonlinear molecular chains, Phys. Scr., 30 (1984), 426. https://doi.org/10.1088/0031-8949/30/6/010

[10] A. V. Zolotaryuk, K. H. Spatschek, A. V. Savin, Supersonic mechanisms for charge and energy transfers in anharmonic molecular chains, Phys. Rev. B, 54 (1996), 266. https://doi.org/10.1103/PhysRevB.54.266

[11] L. A. Cisneros-Ake, J. F. Solano-Peláez, Bright and dark solitons in the unidirectional long wave limit for the energy transfer on anharmonic crystal lattices, Physica D, 346 (2017), 20-27. https://doi.org/10.1016/j.physd.2017.02.001

[12] L. A. Cisneros-Ake, H. P. Prado, D. J. López-Villatoro, R. Carretero-González, Multi-hump bright solitons in a Schrödinger-mKdV system, Phys. Lett. A, 382 (2018), 837-845. https://doi.org/10.1016/j.physleta.2018.01.031

[13] E. Buhe, G. W. Bluman, Symmetry reductions, exact solutions, and conservation laws of the generalized Zakharov equations, J. Math. Phys., 56 (2015), 101501. https://doi.org/10.1063/1.4931962

[14] S. Bilige, T. Chaolu, X. Wang, Application of the extended simplest equation method to the coupled Schrödinger-Boussinesq equation, Appl. Math. Comput., 224 (2013), 517-523. https://doi.org/10.1016/j.amc.2013.08.083

[15] M. A. Abdelkawy, A. H. Bhrawy, E. Zerrad, A. Biswas, Application of tanh method to complex coupled nonlinear evolution equations, Acta Phys. Pol. A, 129 (2016), 278-283.

[16] J. Rao, D. Mihalache, J. He, Y. Cheng, Dynamics of general higher-order rogue waves in the two-component nonlinear Schrödinger equation coupled to the Boussinesq equation, Commun. Nonlinear Sci. Numer. Simul., 110 (2022), 106382. https://doi.org/10.1016/j.cnsns.2022.106382

[17] H. Li, Y. Chen, D. Zhou, Rogue wave patterns of two-component nonlinear Schrödinger equation coupled to the Boussinesq equation, Commun. Nonlinear Sci. Numer. Simul., 140 (2025), 108406. https://doi.org/10.1016/j.cnsns.2024.108406

[18] D. Baleanu, M. Inc, A. I. Aliyu, A. Yusuf, The investigation of soliton solutions and conservation laws to the coupled generalized Schrödinger-Boussinesq system, Waves Random Complex Media, 29 (2019), 77-92. https://doi.org/10.1080/17455030.2017.1412539

[19] L. Huang, D. S. Wang, X. Zhu, Long-time asymptotics of the Tzitzéica equation on the line, Math. Ann., 395 (2026), 18. https://doi.org/10.1007/s00208-026-03445-1

[20] D. S. Wang, X. Zhu, Long-time asymptotics of the Sawada-Kotera equation on the line, SIAM J. Math. Anal., 58 (2026), 3325-3388. https://doi.org/10.1137/25M1777827

[21] D. S. Wang, P. Yan, Rigorous asymptotic analysis for the Riemann problem of the defocusing nonlinear Schrödinger hydrodynamics, Nonlinearity, 38 (2025), 125006. https://doi.org/10.1088/1361-6544/ae1dc4

[22] J. J. Yang, S. F. Tian, Z. Q. Li, The modified Camassa-Holm equation with nonzero background: soliton resolution conjecture and asymptotic stability of N-soliton solutions, Adv. Math., 484 (2026), 110552. https://doi.org/10.1016/j.aim.2025.110552

[23] A. Chatziafratis, T. Ozawa, S. F. Tian, Rigorous analysis of the unified transform method and long-range instabilities for the inhomogeneous time-dependent Schrödinger equation on the quarter-plane, Math. Ann., 389 (2024), 3535-3602. https://doi.org/10.1007/s00208-023-02698-4

[24] P. J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag: New York, 1986.

[25] G. W. Bluman, A. F. Cheviakov, S. C. Anco, Applications of Symmetry Methods to Partial Differential Equations, Springer Applied Mathematics Series 168, Springer: New York, 2010. https://doi.org/10.1007/978-0-387-68028-6

[26] S. C. Anco, Generalization of Noether’s theorem in modern form to non-variational partial differential equations, in Recent progress and Modern Challenges in Applied Mathematics, Modeling and Computational Science, 79 (2017), 119-182. https://doi.org/10.1007/978-1-4939-6969-2_5

[27] S. C. Anco, G. W. Bluman, Direct construction method for conservation laws of partial differential equations Part I: Examples of conservation law classifications, Eur. J. Appl. Math., 13 (2002), 545-566. https://doi.org/10.1017/S095679250100465X

[28] S. C. Anco, G. W. Bluman, Direct construction method for conservation laws of partial differential equations Part II: General treatment, Eur. J. Appl. Math., 13 (2002), 567-585. https://doi.org/10.1017/S0956792501004661

[29] A. R. Chowdhury, N. N. Rao, Painlevé analysis and Bäcklund transformations for coupled generalized Schrödinger-Boussinesq system, Chaos, Solitons Fractals, 9 (1998), 1747-1753. https://doi.org/10.1016/S0960-0779(98)00005-8

[30] M. Abramowitz, I. Stegun, Handbook of Mathematical Functions, Applied Mathematics Series 55, National Bureau of Standards, 1964.

[31] W. Malfliet, W. Hereman, The tanh method: I. Exact solutions of nonlinear evolution and wave equations, Phys. Scr., 54 (1996), 563-568. https://doi.org/10.1088/0031-8949/54/6/003