# American Institute of Mathematical Sciences

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.

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##### 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.
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