# American Institute of Mathematical Sciences

September  2021, 26(9): 5149-5170. doi: 10.3934/dcdsb.2020337

## Recurrent solutions of the Schrödinger-KdV system with boundary forces

 School of Mathematics and Statistics, Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun 130024, China

Received  May 2020 Published  September 2021 Early access  November 2020

Fund Project: This work is supported by NSFC Grant (11701078)

In this paper, we consider the Schrödinger-KdV system with time-dependent boundary external forces. We give conditions on the external forces sufficient for the unique existence of small solutions bounded for all time. Then, we investigate the existence of bounded solution, periodic solution, quasi-periodic solution and almost periodic solution for the Schrödinger-KdV system. The main difficulty is the nonlinear terms in the equations, in order to overcome this difficulty, we establish some properties for the semigroup associated with linear operator which is a crucial tool.

Citation: Mo Chen. Recurrent solutions of the Schrödinger-KdV system with boundary forces. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 5149-5170. doi: 10.3934/dcdsb.2020337
##### References:
 [1] A. Arbieto, A. Corcho and C. Matheus, Rough solutions for the periodic Schrödinger-Korteweg-de Vries system, J. Differential Equations, 230 (2006), 295-336.  doi: 10.1016/j.jde.2006.04.012. [2] P. Amorim and M. Figueira, Convergence of a numerical scheme for a coupled Schrödinger-KdV system, Revista Matemática Complutense, 26 (2013), 409-426.  doi: 10.1007/s13163-012-0097-8. [3] J. Albert and J. A. Pava, Existence and stability of ground-state solutions of a Schrödinger-KdV system, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 133 (2003), 987-1029.  doi: 10.1017/S030821050000278X. [4] D. J. Benney, A general theory for interactions between short and long waves, Studies in Applied Mathematics, 56 (1977), 81-94.  doi: 10.1002/sapm197756181. [5] D. Bekiranov, T. Ogawa and G. Ponce, Weak solvability and well-posedness of a coupled Schrödinger-Korteweg de Vries equation for capillary-gravity wave interactions, Proc. Amer. Math. Soc., 125 (1997), 2907-2919.  doi: 10.1090/S0002-9939-97-03941-5. [6] S. Bhattarai, Solitary waves and a stability analysis of an equation of short and long dispersive waves, Nonlinear Analysis: Theory, Methods & Applications, 75 (2012), 6506-6519.  doi: 10.1016/j.na.2012.07.026. [7] J. Chu, J. M. Coron and P. Shang, Asymptotic stability of a nonlinear Korteweg-de Vries equation with critical lengths, Journal of Differential Equations, 259 (2015), 4045-4085.  doi: 10.1016/j.jde.2015.05.010. [8] V. V. Chepyzhov and M. I. Vishik, Non-autonomous 2D Navier-Stokes system with a simple global attractor and some averaging problems, A tribute to J. L. Lions, ESAIM: Control, Optimisation and Calculus of Variations, 8 (2002), 467-487.  doi: 10.1051/cocv:2002056. [9] V. V. Chepyzhov, M. I. Vishik and W. L. Wendland, On non-autonomous sine-Gordon type equations with a simple global attractor and some averaging, Discrete Contin. Dyn. Syst, 12 (2005), 27-38.  doi: 10.3934/dcds.2005.12.27. [10] V. Chepyzhov and M. Vishik, Attractors for nonautonomous Navier-Stokes system and other partial differential equations, Instability in Models Connected with Fluid Flows I, 6 (2008), 135-265.  doi: 10.1007/978-0-387-75217-4_4. [11] A. J. Corcho F. Linares, Well-posedness for the Schrödinger-Korteweg-de Vries system, Trans. Amer. Math. Soc., 359 (2007), 4089-4106.  doi: 10.1090/S0002-9947-07-04239-0. [12] J. P. Dias, M. Figueira and F. Oliveira, Well-posedness and existence of bound states for a coupled Schrödinger-gKdV system, Nonlinear Analysis: Theory, Methods & Applications, 73 (2010), 2686-2698.  doi: 10.1016/j.na.2010.06.049. [13] P. Gao, Recurrent solutions of the linearly coupled complex cubic-quintic Ginzburg-Landau equations, Mathematical Methods in the Applied Sciences, 41 (2018), 2769-2794.  doi: 10.1002/mma.4778. [14] P. Gao, Recurrent solutions of the derivative Ginzburg-Landau equation with boundary forces, Applicable Analysis, 97 (2018), 2743-2761.  doi: 10.1080/00036811.2017.1387250. [15] P. Gao and Y. Li, Averaging principle for the Schrödinger equations, Discrete & Continuous Dynamical Systems-Series B, 22 (2017), 2147-2168.  doi: 10.3934/dcdsb.2017089. [16] A. Golbabai and A. Safdari-Vaighani, A meshless method for numerical solution of the coupled Schrödinger-KdV equations, Computing, 92 (2011), 225-242.  doi: 10.1007/s00607-010-0138-4. [17] B. Guo and C. Feng-Xin, Finite-dimensional behavior of global attractors for weakly damped and forced kdv equations coupling with nonlinear Schrödinger equations, Nonlinear Analysis: Theory, Methods & Applications, 29 (1997), 569-584.  doi: 10.1016/S0362-546X(96)00061-2. [18] B. Guo and L. Shen, The periodic initial value problem and the initial value problem for the system of KdV equation coupling with nonlinear Schrödinger equations, Proceedings of DD-3 Symposium, Chang Chun, (1982), 417435. [19] Z. Guo and Y. Wang, On the well-posedness of the Schrödinger-Korteweg-de Vries system, Journal of Differential Equations, 249 (2010), 2500-2520.  doi: 10.1016/j.jde.2010.04.016. [20] B. Guo and C. Miao, Well-posedness of the Cauchy problem for the coupled system of the Schrödinger-KdV equations, Acta Mathematica Sinica, 15 (1999), 215-224.  doi: 10.1007/BF02650665. [21] 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.  doi: 10.1143/JPSJ.39.1379. [22] J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, vol.I, Grundlehren Math. Wiss., Band 181, Springer-Verlag, NewYork-Heidelberg, translated fromthe French by P.Kenneth, 1972. [23] V. G. Makhankov, On stationary solutions of the Schrödinger equation with a self-consistent potential satisfying Boussinesq's equation, Physics Letters A, 50 (1974), 42-44.  doi: 10.1016/0375-9601(74)90344-2. [24] K. Nishikawa, H. Hojo, K. Mima et al., Coupled nonlinear electron-plasma and ion-acoustic waves, Physical Review Letters, 33 (1974), 148. doi: 10.1103/PhysRevLett.33.148. [25] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci., vol.44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [26] J. A. Pava, Stability of solitary wave solutions for equations of short and long dispersive waves, Electronic Journal of Differential Equations, 72 (2006), 18p. [27] G. Perla Menzala, C. F. Vasconcellos and E. Zuazua, Stabilization of the Korteweg-de Vries equation with localized damping, Quarterly of Applied Mathematics, 60 (2002), 111-129.  doi: 10.1090/qam/1878262. [28] L. Rosier and B. Y. Zhang, Global stabilization of the generalized Korteweg–de Vries equation posed on a finite domain, SIAM Journal on Control and Optimization, 45 (2006), 927-956.  doi: 10.1137/050631409. [29] L. Rosier, Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain, ESAIM: Control, Optimisation and Calculus of Variations, 2 (1997), 33-55.  doi: 10.1051/cocv:1997102. [30] M. Tsutsumi, Well-posedness of the Cauchy problem for a coupled Schrödinger-KdV equation, Internat. Ser. Math. Sci. Appl., 2 (1993), 513-528. [31] M. Usman and B. Zhang, Forced oscillations of a class of nonlinear dispersive wave equations and their stability, Journal of Systems Science and Complexity, 20 (2007), 284-292.  doi: 10.1007/s11424-007-9025-2. [32] H. Wang and S. Cui, The Cauchy problem for the Schrödinger-KdV system, Journal of Differential Equations, 250 (2011), 3559-3583.  doi: 10.1016/j.jde.2011.02.008. [33] N. Yajima and J. Satsuma, Soliton solutions in a diatomic lattice system, Progress of Theoretical Physics, 62 (1979), 370-378.  doi: 10.1143/PTP.62.370.

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##### References:
 [1] A. Arbieto, A. Corcho and C. Matheus, Rough solutions for the periodic Schrödinger-Korteweg-de Vries system, J. Differential Equations, 230 (2006), 295-336.  doi: 10.1016/j.jde.2006.04.012. [2] P. Amorim and M. Figueira, Convergence of a numerical scheme for a coupled Schrödinger-KdV system, Revista Matemática Complutense, 26 (2013), 409-426.  doi: 10.1007/s13163-012-0097-8. [3] J. Albert and J. A. Pava, Existence and stability of ground-state solutions of a Schrödinger-KdV system, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 133 (2003), 987-1029.  doi: 10.1017/S030821050000278X. [4] D. J. Benney, A general theory for interactions between short and long waves, Studies in Applied Mathematics, 56 (1977), 81-94.  doi: 10.1002/sapm197756181. [5] D. Bekiranov, T. Ogawa and G. Ponce, Weak solvability and well-posedness of a coupled Schrödinger-Korteweg de Vries equation for capillary-gravity wave interactions, Proc. Amer. Math. Soc., 125 (1997), 2907-2919.  doi: 10.1090/S0002-9939-97-03941-5. [6] S. Bhattarai, Solitary waves and a stability analysis of an equation of short and long dispersive waves, Nonlinear Analysis: Theory, Methods & Applications, 75 (2012), 6506-6519.  doi: 10.1016/j.na.2012.07.026. [7] J. Chu, J. M. Coron and P. Shang, Asymptotic stability of a nonlinear Korteweg-de Vries equation with critical lengths, Journal of Differential Equations, 259 (2015), 4045-4085.  doi: 10.1016/j.jde.2015.05.010. [8] V. V. Chepyzhov and M. I. Vishik, Non-autonomous 2D Navier-Stokes system with a simple global attractor and some averaging problems, A tribute to J. L. Lions, ESAIM: Control, Optimisation and Calculus of Variations, 8 (2002), 467-487.  doi: 10.1051/cocv:2002056. [9] V. V. Chepyzhov, M. I. Vishik and W. L. Wendland, On non-autonomous sine-Gordon type equations with a simple global attractor and some averaging, Discrete Contin. Dyn. Syst, 12 (2005), 27-38.  doi: 10.3934/dcds.2005.12.27. [10] V. Chepyzhov and M. Vishik, Attractors for nonautonomous Navier-Stokes system and other partial differential equations, Instability in Models Connected with Fluid Flows I, 6 (2008), 135-265.  doi: 10.1007/978-0-387-75217-4_4. [11] A. J. Corcho F. Linares, Well-posedness for the Schrödinger-Korteweg-de Vries system, Trans. Amer. Math. Soc., 359 (2007), 4089-4106.  doi: 10.1090/S0002-9947-07-04239-0. [12] J. P. Dias, M. Figueira and F. Oliveira, Well-posedness and existence of bound states for a coupled Schrödinger-gKdV system, Nonlinear Analysis: Theory, Methods & Applications, 73 (2010), 2686-2698.  doi: 10.1016/j.na.2010.06.049. [13] P. Gao, Recurrent solutions of the linearly coupled complex cubic-quintic Ginzburg-Landau equations, Mathematical Methods in the Applied Sciences, 41 (2018), 2769-2794.  doi: 10.1002/mma.4778. [14] P. Gao, Recurrent solutions of the derivative Ginzburg-Landau equation with boundary forces, Applicable Analysis, 97 (2018), 2743-2761.  doi: 10.1080/00036811.2017.1387250. [15] P. Gao and Y. Li, Averaging principle for the Schrödinger equations, Discrete & Continuous Dynamical Systems-Series B, 22 (2017), 2147-2168.  doi: 10.3934/dcdsb.2017089. [16] A. Golbabai and A. Safdari-Vaighani, A meshless method for numerical solution of the coupled Schrödinger-KdV equations, Computing, 92 (2011), 225-242.  doi: 10.1007/s00607-010-0138-4. [17] B. Guo and C. Feng-Xin, Finite-dimensional behavior of global attractors for weakly damped and forced kdv equations coupling with nonlinear Schrödinger equations, Nonlinear Analysis: Theory, Methods & Applications, 29 (1997), 569-584.  doi: 10.1016/S0362-546X(96)00061-2. [18] B. Guo and L. Shen, The periodic initial value problem and the initial value problem for the system of KdV equation coupling with nonlinear Schrödinger equations, Proceedings of DD-3 Symposium, Chang Chun, (1982), 417435. [19] Z. Guo and Y. Wang, On the well-posedness of the Schrödinger-Korteweg-de Vries system, Journal of Differential Equations, 249 (2010), 2500-2520.  doi: 10.1016/j.jde.2010.04.016. [20] B. Guo and C. Miao, Well-posedness of the Cauchy problem for the coupled system of the Schrödinger-KdV equations, Acta Mathematica Sinica, 15 (1999), 215-224.  doi: 10.1007/BF02650665. [21] 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.  doi: 10.1143/JPSJ.39.1379. [22] J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, vol.I, Grundlehren Math. Wiss., Band 181, Springer-Verlag, NewYork-Heidelberg, translated fromthe French by P.Kenneth, 1972. [23] V. G. Makhankov, On stationary solutions of the Schrödinger equation with a self-consistent potential satisfying Boussinesq's equation, Physics Letters A, 50 (1974), 42-44.  doi: 10.1016/0375-9601(74)90344-2. [24] K. Nishikawa, H. Hojo, K. Mima et al., Coupled nonlinear electron-plasma and ion-acoustic waves, Physical Review Letters, 33 (1974), 148. doi: 10.1103/PhysRevLett.33.148. [25] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci., vol.44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [26] J. A. Pava, Stability of solitary wave solutions for equations of short and long dispersive waves, Electronic Journal of Differential Equations, 72 (2006), 18p. [27] G. Perla Menzala, C. F. Vasconcellos and E. Zuazua, Stabilization of the Korteweg-de Vries equation with localized damping, Quarterly of Applied Mathematics, 60 (2002), 111-129.  doi: 10.1090/qam/1878262. [28] L. Rosier and B. Y. Zhang, Global stabilization of the generalized Korteweg–de Vries equation posed on a finite domain, SIAM Journal on Control and Optimization, 45 (2006), 927-956.  doi: 10.1137/050631409. [29] L. Rosier, Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain, ESAIM: Control, Optimisation and Calculus of Variations, 2 (1997), 33-55.  doi: 10.1051/cocv:1997102. [30] M. Tsutsumi, Well-posedness of the Cauchy problem for a coupled Schrödinger-KdV equation, Internat. Ser. Math. Sci. Appl., 2 (1993), 513-528. [31] M. Usman and B. Zhang, Forced oscillations of a class of nonlinear dispersive wave equations and their stability, Journal of Systems Science and Complexity, 20 (2007), 284-292.  doi: 10.1007/s11424-007-9025-2. [32] H. Wang and S. Cui, The Cauchy problem for the Schrödinger-KdV system, Journal of Differential Equations, 250 (2011), 3559-3583.  doi: 10.1016/j.jde.2011.02.008. [33] N. Yajima and J. Satsuma, Soliton solutions in a diatomic lattice system, Progress of Theoretical Physics, 62 (1979), 370-378.  doi: 10.1143/PTP.62.370.
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