Journal of University of Chinese Academy of Sciences >
Lie symmetry analysis and exact solutions of two-component Camassa-Holm equation
Received date: 2015-11-16
Revised date: 2015-12-24
Online published: 2016-07-15
Supported by
Supported by National Science Foundation of Shanxi(2014021010-1)
Using the Lie group analysis method, we study the two-component Camassa-Holm equation, which models shallow water waves moving over a linear shear flow.The similarity reductions and exact solutions for the equation are obtained.Then the power series solution are considered by using the power series method.Furthermore, the convergence of the power series solution to the equation is shown.The physical significance of the solutions is considered from the transformation group's point of view.
TIAN Hongxia , GAO Be . Lie symmetry analysis and exact solutions of two-component Camassa-Holm equation[J]. Journal of University of Chinese Academy of Sciences, 2016 , 33(4) : 454 -461 . DOI: 10.7523/j.issn.2095-6134.2016.04.004
[1] Olver P J, Rosenau P.Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support[J].Physical Review E, 1996, 53: 1 900-1 906.
[2] Constantin A, Ivanov R I.On an integrable two-component Camassa-Holm shallow water system[J].Physics Letters A, 2008, 372: 7 129-7 132.
[3] Camassa R, Holm D.An integrable shallow water equation with peaked solitons[J].Phys Rev Lett, 1993, 71: 1 661-1 664.
[4] Constantin A, Lannes D.The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations[J].Arch Ration Mech Anal, 2009, 192: 165-186.
[5] Dullin H R, Gottwald G A, Holm D D.An integrable shallow water equation with linear and nonlinear dispersion[J].Phys Rev Lett, 2001, 87: 4 501-4 504.
[6] Ionescu-Krus D.Variational derivation of the Camassa-Holm shallow water equation[J].J Nonlinear Math Phys, 2007, 14: 303-312.
[7] Ivanov R I.Water waves and integrability[J].Philos Trans R Soc Lond Ser A Math Phys Eng Sci, 2007, 365: 2 267-2 280.
[8] Johnson R S.Camassa-Holm, Korteweg-de Vries and related models for water waves[J].J Fluid Mech, 2002, 457: 63-82.
[9] Constantin A, Strauss W A.Stability of a class of solitary waves in compressible elastic rods[J].Phys Lett A, 2000, 270: 140-148.
[10] Dai H H.Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod[J].Acta Mech, 1998, 127: 193-207.
[11] Constantin A.The Hamiltonian structure of the Camassa-Holm equation[J].Expo Math, 1997, 15: 53-85.
[12] Fokas A, Fuchssteiner B.Symplectic structures, their Bäcklund transformation and hereditary symmetries[J].Phys D, 1981, 4: 47-66.
[13] Constantin A.On the scattering problem for the Camassa-Holm equation[J].Proc R Soc Lond Ser A Math Phys Eng Sci, 2001, 457: 953-970.
[14] Lakshmanan M.Tsunami and nonlinear waves[M].Berlin: Springer, 2007.
[15] Constantin A, Johnson R S.Propagation of very long water waves, with vorticity, over variable depth, with applications to tsunamis[J].Fluid Dynam Res, 2008, 4: 175-211.
[16] Constantin A, Escher J.Global existence and blow-up for a shallow water equation[J].Ann Sc Norm Super Pisa Cl Sci, 1998, 26: 303-328.
[17] Constantin A, Escher J.Well-posedness, global existence and blowup phenomena for a periodic quasi-linear hyperbolic equation[J].Comm Pure Appl Math, 1998, 51: 475-504.
[18] Danchin R.A few remarks on the Camassa-Holm equation[J].Differential Integral Equations, 2001, 14: 953-988.
[19] Escher J, Yin Z.Initial boundary value problems of the Camassa-Holm equation[J].Comm Partial Differential Equations, 2008, 33: 377-395.
[20] Escher J, Yin Z.Initial boundary value problems for nonlinear dispersive wave equations[J].J Funct Anal, 2009, 25: 479-508.
[21] Constantin A.Existence of permanent and breaking waves for a shallow water equation: a geometric approach[J].Ann Inst Fourier (Grenoble), 2000, 50: 321-362.
[22] Constantin A, Escher J.On the blow-up rate and the blow-up of breaking waves for a shallow water equation[J].Math Z, 2000, 233: 75-91.
[23] Escher J, Lechtenfeld O, Yin Z.Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation[J].Discrete Contin Dyn Syst Ser A, 2007, 19: 493-513.
[24] Olver P J.Grauate texts in mathematics[M].New York: Springer, 1993.
[25] Bluman G, Anco S.Symmetry and integration methods for differential equations[M].New York: Springer-Verlag, 2002.
[26] Sinkala W, Leach P, Hara J O.Invariance properties of a general-pricing equation[J].J Differential Equations, 2008, 244: 2 820-2 835.
[27] Craddock M, Lennox K.Lie group symmetries as integral transforms of fundamental solutions[J].J Differential Equations, 2007, 232: 652-674.
[28] Liu H, Li J, Liu L.Conservation law classification and integrability of generalized nonlinear second-order equation[J].Commun Theor Phys, 2011, 56: 987-991.
[29] Asmar N H.Partial differential equations with fourier series and boundary value problems[M].2nd ed.Beijing: China Machine Press, 2005.
[30] Rudin W.Principles of mathematical analysis[M].Beijing: China Machine Press, 2004.
/
| 〈 |
|
〉 |