December  2016, 9(6): 1687-1699. doi: 10.3934/dcdss.2016070

Global existence and uniqueness of the solution for the fractional Schrödinger-KdV-Burgers system

1. 

Department of Mathematics, China University of Mining and Technology Beijing, Beijing 100083, China

2. 

Department of Mathematical science, Tsinghua University, Beijing 100084, China

3. 

Institute of Applied Physics and Computational Mathematics, Beijing 100088, China

Received  May 2015 Revised  September 2016 Published  November 2016

In this paper, we consider a fractional Schrödinger-KdV-Burgers system. First, the local existence and uniqueness of solution is obtained by contraction method. Then by some a priori estimates, global existence and uniqueness of smooth solution for this system is proved. Moreover, the regularity of the solution is improved.
Citation: Chunxiao Guo, Fan Cui, Yongqian Han. Global existence and uniqueness of the solution for the fractional Schrödinger-KdV-Burgers system. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 1687-1699. doi: 10.3934/dcdss.2016070
References:
[1]

D. Bekiranov, T. Ogawa and G. Ponce, Weak solvability and well-posedness of a couples Schrödinger-Korteweg de Vries equation for capillary-gravity wave interactions, Processdings of the AMS, 125 (1997), 2907-2919. doi: 10.1090/S0002-9939-97-03941-5.

[2]

J. Canosa and J. Gazdag, The Korteweg-de Vries-Burgers equation, Journal of Computational Physics, 23 (1977), 393-403. doi: 10.1016/0021-9991(77)90070-5.

[3]

G. Carlson, Investigation of Fractional Capacitor Approximations by Means of Regular Newton Processes, Kansas State University, 1964.

[4]

R. R. Coifman and Y. Meyer, Nonlinear harmonic analysis, operator theory and P.D.E., Beijing Lectures in Harmonic Analysis, Ann. of Math. Stud., Princeton Univ. Press, Princeton, NJ, 112 (1986), 3-45.

[5]

A. J. Corcho and 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.

[6]

W. Deng, Generalized synchronization in fractional order systems, Physical Review E, 75 (2007), 056201. doi: 10.1103/PhysRevE.75.056201.

[7]

A. Friedman, Partial Differential Equations, Holt, Reinhart and Winston, 1969.

[8]

B. Guo, The initial and periodic value problems of one class couples Schrödinger-Korteweg-de Vries equations, Acta Math. Sinica, Chinese Series, 26 (1983), 513-532.

[9]

B. Guo, Y. Han and J. Xin, Existence of the global smooth solution to the period boundary value problem of fractional nonlinear Schrödinger equation, Appl. Math. and Comp., 204 (2008), 468-477. doi: 10.1016/j.amc.2008.07.003.

[10]

B. Guo and C. Miao, Well-posedness of the Cauchy problem for the coupled system of the Schrödinger-KdV equations, Acta Math. Sinica, English Series, 15 (1999), 215-224. doi: 10.1007/BF02650665.

[11]

X. Guo and M. Xu, Some physical applications of fractional Schrödinger equation, J. Math. Phys., 47 (2006), 082104, 9pp. doi: 10.1063/1.2235026.

[12]

R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. doi: 10.1142/9789812817747.

[13]

T. Kato, Liapunov functions and monotonicity in the Navier-Stokes equations, Lecture Notes in Mathematics, Springer-Verlag, 1450 (1990), 53-63. doi: 10.1007/BFb0084898.

[14]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907. doi: 10.1002/cpa.3160410704.

[15]

C. Kenig, G. Ponce and L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc., 4 (1991), 323-347. doi: 10.1090/S0894-0347-1991-1086966-0.

[16]

C. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure. Appl. Math., 46 (1993), 527-620. doi: 10.1002/cpa.3160460405.

[17]

D. Kusnezov, A. Bulgac and G. Dang, Quantum levy processes and fractional kinetics, Physical Review Letters, 82 (1999), 1136-1139. doi: 10.1103/PhysRevLett.82.1136.

[18]

N. Laskin, Fractional quantum mechanics and Lévy integrals, Phys. Lett. A, 268 (2000), 298-305. doi: 10.1016/S0375-9601(00)00201-2.

[19]

N. Laskin, Fractional quantum mechanics, Phys. Rev. E, 62 (2000), 3135-3145. doi: 10.1103/PhysRevE.62.3135.

[20]

N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 056108, 7pp. doi: 10.1103/PhysRevE.66.056108.

[21]

F. Mainardi, Fractional calculus: Some basic problems in continuum and statistical of the second kind, Math. Comp., 45 (1985), 463-469.

[22]

F. Mainardi, Fractional relaxation-oscillation and fractional diffusion-wave phenomena, Chaos Solitons Fractals, 7 (1996), 1461-1477. doi: 10.1016/0960-0779(95)00125-5.

[23]

K. Nishihara and S. V. Rajopadhye, Asymptotic behavior of solutions to the Korteweg-de Vries-Burgers equation, Diff. Int. Equation, 11 (1998), 85-93.

[24]

A. Oustaloup and P. Coiffet, Systemes Asservis Lineaires D'ordre Fractionnaire: Theorie et Pratique, Masson, 1983.

[25]

I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Deriva Tives, Fractional Differential Equations, to Methods of Their Solution and some of Their Applications, Academic Press, San Diego, 1999.

[26]

N. Sugimoto, Burgers equation with a fractional derivative; hereditary effects on nonlinear acoustic waves, Journal of Fluid Mechanics Digital Archive, 225 (1991), 631-653. doi: 10.1017/S0022112091002203.

[27]

D. Tomasz and C. Sun, Asymptotic behavior of the generalized Korteweg-de Vries-Burgers equation, J. Evol. Equ., 10 (2010), 571-595. doi: 10.1007/s00028-010-0062-2.

[28]

B. J. West, M. Bologna and P. Grigolini, Physical of Fractal Operators, Springer, New York, 2003. doi: 10.1007/978-0-387-21746-8.

[29]

H. Yin, H. Zhao and L. Zhou, Convergence rate of solutions toward traveling waves for the Cauchy problem of generalized Korteweg-de Vries-Burgers equations, Nonlinear Anal. TMA, 71 (2009), 3981-3991. doi: 10.1016/j.na.2009.02.068.

show all references

References:
[1]

D. Bekiranov, T. Ogawa and G. Ponce, Weak solvability and well-posedness of a couples Schrödinger-Korteweg de Vries equation for capillary-gravity wave interactions, Processdings of the AMS, 125 (1997), 2907-2919. doi: 10.1090/S0002-9939-97-03941-5.

[2]

J. Canosa and J. Gazdag, The Korteweg-de Vries-Burgers equation, Journal of Computational Physics, 23 (1977), 393-403. doi: 10.1016/0021-9991(77)90070-5.

[3]

G. Carlson, Investigation of Fractional Capacitor Approximations by Means of Regular Newton Processes, Kansas State University, 1964.

[4]

R. R. Coifman and Y. Meyer, Nonlinear harmonic analysis, operator theory and P.D.E., Beijing Lectures in Harmonic Analysis, Ann. of Math. Stud., Princeton Univ. Press, Princeton, NJ, 112 (1986), 3-45.

[5]

A. J. Corcho and 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.

[6]

W. Deng, Generalized synchronization in fractional order systems, Physical Review E, 75 (2007), 056201. doi: 10.1103/PhysRevE.75.056201.

[7]

A. Friedman, Partial Differential Equations, Holt, Reinhart and Winston, 1969.

[8]

B. Guo, The initial and periodic value problems of one class couples Schrödinger-Korteweg-de Vries equations, Acta Math. Sinica, Chinese Series, 26 (1983), 513-532.

[9]

B. Guo, Y. Han and J. Xin, Existence of the global smooth solution to the period boundary value problem of fractional nonlinear Schrödinger equation, Appl. Math. and Comp., 204 (2008), 468-477. doi: 10.1016/j.amc.2008.07.003.

[10]

B. Guo and C. Miao, Well-posedness of the Cauchy problem for the coupled system of the Schrödinger-KdV equations, Acta Math. Sinica, English Series, 15 (1999), 215-224. doi: 10.1007/BF02650665.

[11]

X. Guo and M. Xu, Some physical applications of fractional Schrödinger equation, J. Math. Phys., 47 (2006), 082104, 9pp. doi: 10.1063/1.2235026.

[12]

R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. doi: 10.1142/9789812817747.

[13]

T. Kato, Liapunov functions and monotonicity in the Navier-Stokes equations, Lecture Notes in Mathematics, Springer-Verlag, 1450 (1990), 53-63. doi: 10.1007/BFb0084898.

[14]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907. doi: 10.1002/cpa.3160410704.

[15]

C. Kenig, G. Ponce and L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc., 4 (1991), 323-347. doi: 10.1090/S0894-0347-1991-1086966-0.

[16]

C. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure. Appl. Math., 46 (1993), 527-620. doi: 10.1002/cpa.3160460405.

[17]

D. Kusnezov, A. Bulgac and G. Dang, Quantum levy processes and fractional kinetics, Physical Review Letters, 82 (1999), 1136-1139. doi: 10.1103/PhysRevLett.82.1136.

[18]

N. Laskin, Fractional quantum mechanics and Lévy integrals, Phys. Lett. A, 268 (2000), 298-305. doi: 10.1016/S0375-9601(00)00201-2.

[19]

N. Laskin, Fractional quantum mechanics, Phys. Rev. E, 62 (2000), 3135-3145. doi: 10.1103/PhysRevE.62.3135.

[20]

N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 056108, 7pp. doi: 10.1103/PhysRevE.66.056108.

[21]

F. Mainardi, Fractional calculus: Some basic problems in continuum and statistical of the second kind, Math. Comp., 45 (1985), 463-469.

[22]

F. Mainardi, Fractional relaxation-oscillation and fractional diffusion-wave phenomena, Chaos Solitons Fractals, 7 (1996), 1461-1477. doi: 10.1016/0960-0779(95)00125-5.

[23]

K. Nishihara and S. V. Rajopadhye, Asymptotic behavior of solutions to the Korteweg-de Vries-Burgers equation, Diff. Int. Equation, 11 (1998), 85-93.

[24]

A. Oustaloup and P. Coiffet, Systemes Asservis Lineaires D'ordre Fractionnaire: Theorie et Pratique, Masson, 1983.

[25]

I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Deriva Tives, Fractional Differential Equations, to Methods of Their Solution and some of Their Applications, Academic Press, San Diego, 1999.

[26]

N. Sugimoto, Burgers equation with a fractional derivative; hereditary effects on nonlinear acoustic waves, Journal of Fluid Mechanics Digital Archive, 225 (1991), 631-653. doi: 10.1017/S0022112091002203.

[27]

D. Tomasz and C. Sun, Asymptotic behavior of the generalized Korteweg-de Vries-Burgers equation, J. Evol. Equ., 10 (2010), 571-595. doi: 10.1007/s00028-010-0062-2.

[28]

B. J. West, M. Bologna and P. Grigolini, Physical of Fractal Operators, Springer, New York, 2003. doi: 10.1007/978-0-387-21746-8.

[29]

H. Yin, H. Zhao and L. Zhou, Convergence rate of solutions toward traveling waves for the Cauchy problem of generalized Korteweg-de Vries-Burgers equations, Nonlinear Anal. TMA, 71 (2009), 3981-3991. doi: 10.1016/j.na.2009.02.068.

[1]

Jiaxiang Cai, Juan Chen, Bin Yang. Fully decoupled schemes for the coupled Schrödinger-KdV system. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5523-5538. doi: 10.3934/dcdsb.2019069

[2]

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

[3]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247

[4]

Gerd Grubb. Limited regularity of solutions to fractional heat and Schrödinger equations. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3609-3634. doi: 10.3934/dcds.2019148

[5]

Wenxiong Chen, Congming Li. A priori estimate for the Nirenberg problem. Discrete and Continuous Dynamical Systems - S, 2008, 1 (2) : 225-233. doi: 10.3934/dcdss.2008.1.225

[6]

Mengyao Chen, Qi Li, Shuangjie Peng. Bound states for fractional Schrödinger-Poisson system with critical exponent. Discrete and Continuous Dynamical Systems - S, 2021, 14 (6) : 1819-1835. doi: 10.3934/dcdss.2021038

[7]

Yongsheng Jiang, Huan-Song Zhou. A sharp decay estimate for nonlinear Schrödinger equations with vanishing potentials. Communications on Pure and Applied Analysis, 2010, 9 (6) : 1723-1730. doi: 10.3934/cpaa.2010.9.1723

[8]

Hiroyuki Hirayama. Well-posedness and scattering for a system of quadratic derivative nonlinear Schrödinger equations with low regularity initial data. Communications on Pure and Applied Analysis, 2014, 13 (4) : 1563-1591. doi: 10.3934/cpaa.2014.13.1563

[9]

Hartmut Pecher. Low regularity well-posedness for the 3D Klein - Gordon - Schrödinger system. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1081-1096. doi: 10.3934/cpaa.2012.11.1081

[10]

Salah Missaoui, Ezzeddine Zahrouni. Regularity of the attractor for a coupled Klein-Gordon-Schrödinger system with cubic nonlinearities in $\mathbb{R}^2$. Communications on Pure and Applied Analysis, 2015, 14 (2) : 695-716. doi: 10.3934/cpaa.2015.14.695

[11]

Masoumeh Hosseininia, Mohammad Hossein Heydari, Carlo Cattani. A wavelet method for nonlinear variable-order time fractional 2D Schrödinger equation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2273-2295. doi: 10.3934/dcdss.2020295

[12]

Giuseppe Floridia, Hiroshi Takase, Masahiro Yamamoto. A Carleman estimate and an energy method for a first-order symmetric hyperbolic system. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022016

[13]

Mouhamed Moustapha Fall. Regularity estimates for nonlocal Schrödinger equations. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1405-1456. doi: 10.3934/dcds.2019061

[14]

Fengping Yao. Optimal regularity for parabolic Schrödinger operators. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1407-1414. doi: 10.3934/cpaa.2013.12.1407

[15]

Salah Missaoui. Regularity of the attractor for a coupled Klein-Gordon-Schrödinger system in $ \mathbb{R}^3 $ nonlinear KGS system. Communications on Pure and Applied Analysis, 2022, 21 (2) : 567-584. doi: 10.3934/cpaa.2021189

[16]

Guowei Dai, Rushun Tian, Zhitao Zhang. Global bifurcations and a priori bounds of positive solutions for coupled nonlinear Schrödinger Systems. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 1905-1927. doi: 10.3934/dcdss.2019125

[17]

Xia Sun, Kaimin Teng. Positive bound states for fractional Schrödinger-Poisson system with critical exponent. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3735-3768. doi: 10.3934/cpaa.2020165

[18]

Kaimin Teng, Xian Wu. Concentration of bound states for fractional Schrödinger-Poisson system via penalization methods. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1157-1187. doi: 10.3934/cpaa.2022014

[19]

Weizhu Bao, Chunmei Su. Uniform error estimates of a finite difference method for the Klein-Gordon-Schrödinger system in the nonrelativistic and massless limit regimes. Kinetic and Related Models, 2018, 11 (4) : 1037-1062. doi: 10.3934/krm.2018040

[20]

E. Compaan, N. Tzirakis. Low-regularity global well-posedness for the Klein-Gordon-Schrödinger system on $ \mathbb{R}^+ $. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 3867-3895. doi: 10.3934/dcds.2019156

2020 Impact Factor: 2.425

Metrics

  • PDF downloads (100)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]