Abstract
In this paper, we study the coupled nonlinear Schrödinger–Korteweg–de Vries system with periodic potential. By using the variational method and Nehari manifold, we obtain the existence of non-trivial ground state solution.
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Ardila, A.H.: Existence and stability of a two-parameter family of solitary waves for a logarithmic NLS–KdV system. Nonlinear Anal. 189, 111563 (2019)
Bi, W.-J., Tang, C.-L.: Ground state solutions for a non-autonomous nonlinear Schrödinger–KdV system. Front. Math. China 15, 851–866 (2020)
Bartsch, T., Ding, Y.: On a nonlinear Schrödinger equation with periodic potential. Math. Ann. 313, 15–37 (1999)
Colorado, E.: Existence of bound and ground states for a system of coupled nonlinear Schrödinger–KdV equations. C. R. Math. Acad. Sci. Paris 353, 511–516 (2015)
Colorado, E.: On the existence of bound and ground states for some coupled nonlinear Schrödinger–Korteweg–de Vries equations. Adv. Nonlinear Anal. 6, 407–426 (2017)
Coti Zelati, V., Rabinowitz, P.H.: Homoclinic type solutions for a semilinear elliptic PDE on \({\mathbb{R}}^{N}\). Commun. Pure Appl. Math. 45, 1217–1269 (1992)
Dias, J.-P., Mário, F., Filipe, O.: Existence of bound states for the coupled Schrödinger–KdV system with cubic nonlinearity. C. R. Math. Acad. Sci. Paris 348, 1079–1082 (2010)
Ekeland, I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974)
Kawahara, T., Sugimoto, N., Kakutani, T.: Nonlinear interaction between short and long capillary-gravity waves. J. Phys. Soc. Jpn. 39, 1379–1386 (1975)
Kryszewski, W., Szulkin, A.: Generalized linking theorem with an application to semilinear Schrödinger equation. Adv. Differ. Equ. 3, 441–472 (1998)
Liu, C.-G., Zheng, Y.-Q.: On soliton solutions to a class of Schrödinger–KdV systems. Proc. Am. Math. Soc. 141, 3477–3484 (2013)
Liao, F., Zhang, L.-M.: High accuracy split-step finite difference method for Schrödinger–KdV equations. Commun. Theort. Phys. 70, 413–422 (2018)
Liang, F.-F., Wu, X.-P., Tang, C.-L.: Normalized Ground-State Solution for the Schrödinger–KdV System. Mediterr. J. Math. 19, 1–15 (2022)
Liu, J., Liao, J.-F., Tang, C.-L.: A positive ground state solution for a class of asymptotically periodic Schrödinger equations. Comput. Math. Appl. 71, 965–976 (2016)
Willem, M.: Minimax Theorems. Birkhäuser, Basel (1996)
Pankov, A.: Periodic nonlinear Schrödinger equation with application to photonic crystals. Milan J. Math. 73, 259–287 (2005)
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Liang, FF., Wu, XP. & Tang, CL. Ground State Solution for Schrödinger–KdV System with Periodic Potential. Qual. Theory Dyn. Syst. 22, 39 (2023). https://doi.org/10.1007/s12346-023-00741-y
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DOI: https://doi.org/10.1007/s12346-023-00741-y