On soliton solutions to a class of Schrödinger-K$\mathbf {d}$V systems
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- by Chungen Liu and Youquan Zheng PDF
- Proc. Amer. Math. Soc. 141 (2013), 3477-3484 Request permission
Abstract:
In this paper, we consider a class of coupled nonlinear Schrödinger-KdV systems in the whole space via the Nehari manifold method. The existence of nontrivial solutions with both of the components nonzero is obtained.References
- Antonio Ambrosetti and Eduardo Colorado, Standing waves of some coupled nonlinear Schrödinger equations, J. Lond. Math. Soc. (2) 75 (2007), no. 1, 67–82. MR 2302730, DOI 10.1112/jlms/jdl020
- John Albert and Jaime Angulo Pava, Existence and stability of ground-state solutions of a Schrödinger-KdV system, Proc. Roy. Soc. Edinburgh Sect. A 133 (2003), no. 5, 987–1029. MR 2018323, DOI 10.1017/S030821050000278X
- João-Paulo Dias, Mário Figueira, and Filipe Oliveira, Existence of bound states for the coupled Schrödinger-KdV system with cubic nonlinearity, C. R. Math. Acad. Sci. Paris 348 (2010), no. 19-20, 1079–1082 (English, with English and French summaries). MR 2735011, DOI 10.1016/j.crma.2010.09.018
- João-Paulo Dias, Mário Figueira, and Filipe Oliveira, Well-posedness and existence of bound states for a coupled Schrödinger-gKdV system, Nonlinear Anal. 73 (2010), no. 8, 2686–2698. MR 2674102, DOI 10.1016/j.na.2010.06.049
- I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974), 324–353. MR 346619, DOI 10.1016/0022-247X(74)90025-0
- Bo Ling Guo, Existence and uniqueness of the global solution of the Cauchy problem and the periodic initial value problem for a class of coupled systems of KdV-nonlinear Schrödinger equations, Acta Math. Sinica 26 (1983), no. 5, 513–532 (Chinese). MR 747175
- Bo Ling Guo and Ya Ping Wu, Orbital stability of solitary waves for the nonlinear derivative Schrödinger equation, J. Differential Equations 123 (1995), no. 1, 35–55. MR 1359911, DOI 10.1006/jdeq.1995.1156
- Man Kam Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $\textbf {R}^n$, Arch. Rational Mech. Anal. 105 (1989), no. 3, 243–266. MR 969899, DOI 10.1007/BF00251502
- T. Kawahara, N. Sugimoto and T. Kakutani, Nonlinear interaction between short and long capillary-gravity waves, J. Phys. Soc. Jpn., 39(1975), 1379–1386.
- Lin Chen, Orbital stability of solitary waves of the nonlinear Schrödinger-KdV equation, J. Partial Differential Equations 12 (1999), no. 1, 11–25. MR 1681850
- V. Makhankov, On stationary solutions of the Schrödinger equation with a self-consistent potential satisfying Boussinesq’s equation, Phys. Lett. A, 50(1974), 42–44.
- L. A. Maia, E. Montefusco, and B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Differential Equations 229 (2006), no. 2, 743–767. MR 2263573, DOI 10.1016/j.jde.2006.07.002
- L. A. Maia, E. Montefusco, and B. Pellacci, Infinitely many nodal solutions for a weakly coupled nonlinear Schrödinger system, Commun. Contemp. Math. 10 (2008), no. 5, 651–669. MR 2446894, DOI 10.1142/S0219199708002934
- K. Nishikawa, H. Hojo, K. Mima and H. Ikezi, Coupled nonlinear electron-plasma and ion-acoustic waves, Phys. Rev. Lett., 33(1974), 148–151.
- Wei-Ming Ni and Izumi Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Math. J. 70 (1993), no. 2, 247–281. MR 1219814, DOI 10.1215/S0012-7094-93-07004-4
- Jaime Angulo and J. Fabio Montenegro, Existence and evenness of solitary-wave solutions for an equation of short and long dispersive waves, Nonlinearity 13 (2000), no. 5, 1595–1611. MR 1781810, DOI 10.1088/0951-7715/13/5/310
- Susanna Terracini and Gianmaria Verzini, Solutions of prescribed number of zeroes to a class of superlinear ODE’s systems, NoDEA Nonlinear Differential Equations Appl. 8 (2001), no. 3, 323–341. MR 1841262, DOI 10.1007/PL00001451
- Michel Willem, Minimax theorems, Progress in Nonlinear Differential Equations and their Applications, vol. 24, Birkhäuser Boston, Inc., Boston, MA, 1996. MR 1400007, DOI 10.1007/978-1-4612-4146-1
- N. Yajima and J. Satsuma, Soliton solutions in a diatomic lattice system, Prog. Theor. Phys., 62(1979), 370–378.
Additional Information
- Chungen Liu
- Affiliation: School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, People’s Republic of China
- Email: liucg@nankai.edu.cn
- Youquan Zheng
- Affiliation: School of Mathematical Sciences, Nankai University, Tianjin 300071, People’s Republic of China
- Email: zhengyq@mail.nankai.edu.cn
- Received by editor(s): March 30, 2011
- Received by editor(s) in revised form: December 11, 2011
- Published electronically: June 12, 2013
- Additional Notes: The first author was partially supported by NFSC (11071127, 10621101), the 973 Program of STM of China (2011CB808002) and SRFDP
- Communicated by: Yingfei Yi
- © Copyright 2013 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 141 (2013), 3477-3484
- MSC (2010): Primary 35J10, 35J50, 35J60
- DOI: https://doi.org/10.1090/S0002-9939-2013-11629-1
- MathSciNet review: 3080170