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

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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

Chunxiao Guo ^1, , Fan Cui ^2, and Yongqian Han ^3,

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

Abstract
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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.

Keywords: Fractional Schrödinger-KdV-Burgers system, a priori estimate, regularity., contraction method.

Mathematics Subject Classification: Primary: 26A33; Secondary: 35Q53, 35Q5.

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.

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