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  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 & Continuous Dynamical Systems - B, 2021, 26 (9) : 5149-5170. doi: 10.3934/dcdsb.2020337
References:
[1]

A. ArbietoA. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

D. BekiranovT. 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.  Google Scholar

[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.  Google Scholar

[7]

J. ChuJ. 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.  Google Scholar

[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.  Google Scholar

[9]

V. V. ChepyzhovM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

J. P. DiasM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

T. KawaharaN. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

G. Perla MenzalaC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

M. Tsutsumi, Well-posedness of the Cauchy problem for a coupled Schrödinger-KdV equation, Internat. Ser. Math. Sci. Appl., 2 (1993), 513-528.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

A. ArbietoA. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

D. BekiranovT. 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.  Google Scholar

[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.  Google Scholar

[7]

J. ChuJ. 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.  Google Scholar

[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.  Google Scholar

[9]

V. V. ChepyzhovM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

J. P. DiasM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

T. KawaharaN. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

G. Perla MenzalaC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

M. Tsutsumi, Well-posedness of the Cauchy problem for a coupled Schrödinger-KdV equation, Internat. Ser. Math. Sci. Appl., 2 (1993), 513-528.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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