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 & 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. doi: 10.1090/S0002-9939-97-03941-5. Google Scholar

[2]

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

[3]

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

[4]

R. R. Coifman and Y. Meyer, Nonlinear harmonic analysis, operator theory and P.D.E.,, Beijing Lectures in Harmonic Analysis, 112 (1986), 3. Google Scholar

[5]

A. J. Corcho and F. Linares, Well-posedness for the Schrödinger-Korteweg-de Vries system,, Trans. Amer. Math. Soc., 359 (2007), 4089. doi: 10.1090/S0002-9947-07-04239-0. Google Scholar

[6]

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

[7]

A. Friedman, Partial Differential Equations,, Holt, (1969). Google Scholar

[8]

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

[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. doi: 10.1016/j.amc.2008.07.003. Google Scholar

[10]

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

[11]

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

[12]

R. Hilfer, Applications of Fractional Calculus in Physics,, World Scientific, (2000). doi: 10.1142/9789812817747. Google Scholar

[13]

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

[14]

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

[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. doi: 10.1090/S0894-0347-1991-1086966-0. Google Scholar

[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. doi: 10.1002/cpa.3160460405. Google Scholar

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[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, (1999). Google Scholar

[26]

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

[27]

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

[28]

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

[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. doi: 10.1016/j.na.2009.02.068. Google Scholar

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. doi: 10.1090/S0002-9939-97-03941-5. Google Scholar

[2]

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

[3]

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

[4]

R. R. Coifman and Y. Meyer, Nonlinear harmonic analysis, operator theory and P.D.E.,, Beijing Lectures in Harmonic Analysis, 112 (1986), 3. Google Scholar

[5]

A. J. Corcho and F. Linares, Well-posedness for the Schrödinger-Korteweg-de Vries system,, Trans. Amer. Math. Soc., 359 (2007), 4089. doi: 10.1090/S0002-9947-07-04239-0. Google Scholar

[6]

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

[7]

A. Friedman, Partial Differential Equations,, Holt, (1969). Google Scholar

[8]

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

[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. doi: 10.1016/j.amc.2008.07.003. Google Scholar

[10]

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

[11]

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

[12]

R. Hilfer, Applications of Fractional Calculus in Physics,, World Scientific, (2000). doi: 10.1142/9789812817747. Google Scholar

[13]

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

[14]

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

[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. doi: 10.1090/S0894-0347-1991-1086966-0. Google Scholar

[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. doi: 10.1002/cpa.3160460405. Google Scholar

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[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, (1999). Google Scholar

[26]

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

[27]

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

[28]

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

[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. doi: 10.1016/j.na.2009.02.068. Google Scholar

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