On soliton solutions to a class of Schrödinger-K V systems

Authors:
Chungen Liu and Youquan Zheng

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

Published electronically:
June 12, 2013

MathSciNet review:
3080170

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**1.**A. Ambrosetti and E. Colorado,*Standing waves of some coupled nonlinear Schrödinger equations*, J. London Math. Soc., 75(2)(2007), 67-82. MR**2302730 (2008f:35369)****2.**J. Albert and J. Angulo Pava,*Existence and stability of ground-state solutions of a Schrödinger-KdV system*, Proceedings of the Royal Society of Edinburgh, 133A(2003), 987-1029. MR**2018323 (2005f:35269)****3.**J. Dias, M. Figueira and F. Oliveira,*Existence of bound states for the coupled Schrödinger-KdV system with cubic nonlinearity*, Comptes Rendus Mathematique, 348(2010), 19-20, 1079-1082. MR**2735011 (2011g:35348)****4.**J. Dias, M. Figueira and F. Oliveira,*Well-posedness and existence of bound states for a coupled Schrödinger-gKdV system*, Nonlinear Analysis, 73 (2010), 2686-2698. MR**2674102 (2011g:35371)****5.**I. Ekeland,*On the variational principle*, Journal of Mathematical Analysis and Applications, 47(1974), 324-353. MR**0346619 (49:11344)****6.**B. 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), 513-532 (Chinese). MR**747175 (86e:35132)****7.**B. Guo and Y. Wu,*Orbital stability of solitary waves for the nonlinear derivative Schrödinger equation*, J. Diff. Eqns., 123(1995), 35-55. MR**1359911 (96k:35166)****8.**M. K. Kwong,*Uniqueness of positive solutions of in*, Arch. Rat. Mech. Anal., 105(1989), 243-266. MR**969899 (90d:35015)****9.**T. Kawahara, N. Sugimoto and T. Kakutani,*Nonlinear interaction between short and long capillary-gravity waves*, J. Phys. Soc. Jpn., 39(1975), 1379-1386.**10.**L. Chen,*Orbital stability of solitary waves of the nonlinear Schrödinger-KdV equation*, Partial Diff. Eqns., 12(1999), 11-25. MR**1681850 (2000d:35212)****11.**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.**12.**L. A. Maia, E. Montefusco and B. Pellacci,*Positive solutions of a weakly coupled nonlinear Schrödinger system*, J. Diff. Equations, 229(2006), 743-767. MR**2263573 (2007h:35070)****13.**L. A. Maia, E. Montefusco and B. Pellacci,*Infinitely many nodal solutions for a weakly coupled nonlinear Schrödinger system*, Communications in Contemporary Mathematics, 10(5)(2008), 651-669. MR**2446894 (2009m:35108)****14.**K. Nishikawa, H. Hojo, K. Mima and H. Ikezi,*Coupled nonlinear electron-plasma and ion-acoustic waves*, Phys. Rev. Lett., 33(1974), 148-151.**15.**W. Ni and I. Takagi,*Locating the peaks of least energy solutions to a semilinear Neumann problem*, Duke Math. J., 70(1993), 247-281. MR**1219814 (94h:35072)****16.**J. Angulo and J. F. Montenegro,*Existence and evenness of solitary-wave solutions for an equation of short and long dispersive waves*, Nonlinearity, 13(2000), 1595-1611. MR**1781810 (2001e:35142)****17.**S. Terracini and G. Verzini,*Solutions of prescribed number of zeros to a class of superlinear ODE's systems*, Nonlinear Differential Equations Appl., 8(2001), 323-341. MR**1841262 (2002f:34038)****18.**M. Willem,*Minimax Theorems*, Birkhäuser, Boston, 1996. MR**1400007 (97h:58037)****19.**N. Yajima and J. Satsuma,*Soliton solutions in a diatomic lattice system*, Prog. Theor. Phys., 62(1979), 370-378.

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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

DOI:
https://doi.org/10.1090/S0002-9939-2013-11629-1

Keywords:
Schr\"odinger-KdV systems,
Nehari manifold,
nontrivial solution

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

Article copyright:
© Copyright 2013
American Mathematical Society