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Lie symmetries and exact solutions of variable coefficient mKdV equations: an equivalence based approach. (English) Zbl 1245.35114
Summary: Group classification of classes of mKdV-like equations with time-dependent coefficients is carried out. The usage of equivalence transformations appears to be a crucial point for the exhaustive solution of the problem. We prove that all the classes under consideration are normalized. This allows us to formulate the classification results in three ways: up to two kinds of equivalence (which are generated by transformations from the corresponding equivalence groups and all admissible point transformations) and using no equivalence. A simple way for the construction of exact solutions of mKdV-like equations using equivalence transformations is described.
MSC:
35Q53KdV-like (Korteweg-de Vries) equations
35A22Transform methods (PDE)
35B06Symmetries, invariants, etc. (PDE)
References:
[1]Ablowitz, M. J.; Satsuma, J.: Solitons and rational solutions of nonlinear evolution equations, J math phys 19, 2180-2186 (1978) · Zbl 0418.35022 · doi:10.1063/1.523550
[2]Ablowitz, M. J.; Segur, H.: Solitons and inverse scattering transform, (1981)
[3]Bluman, G. W.; Kumei, S.: Symmetries and differential equations, (1989)
[4]Bluman, G. W.; Reid, G. J.; Kumei, S.: New classes of symmetries for partial differential equations, J math phys 29, 806-811 (1988) · Zbl 0669.58037 · doi:10.1063/1.527974
[5]Dos Santos Cardoso-Bihlo, E.; Bihlo, A.; Popovych, R. O.: Enhanced preliminary group classification of a class of generalized diffusion equations, Commun nonlinear sci numer simulat 16, 3622-3638 (2011) · Zbl 1222.35012 · doi:10.1016/j.cnsns.2011.01.011
[6]Güngör, F.; Lahno, V. I.; Zhdanov, R. Z.: Symmetry classification of KdV-type nonlinear evolution equations, J math phys 45, 2280-2313 (2004) · Zbl 1071.35112 · doi:10.1063/1.1737811
[7]Ivanova, N. M.; Popovych, R. O.; Sophocleous, C.: Group analysis of variable coefficient diffusion – convection equations. I. enhanced group classification, Lobachevskii J math 31, 100-122 (2010)
[8]Ivanova NM, Popovych RO, Sophocleous C. Group analysis of variable coefficient diffusion – convection equations. II. Contractions and Exact Solutions. 19 p. Available from: lt;arXiv:0710.3049gt;.
[9]Johnpillai, A. G.; Khalique, C. M.: Lie group classification and invariant solutions of mkdv equation with time-dependent coefficients, Commun nonlinear sci numer simulat 16, 1207-1215 (2011) · Zbl 1221.35338 · doi:10.1016/j.cnsns.2010.06.025
[10]Kingston, J. G.: On point transformation of evolution equations, J phys A: math gen 24, L769-L774 (1991)
[11]Kingston, J. G.; Sophocleous, C.: On form-preserving point transformations of partial differential equations, J phys A: math gen 31, 1597-1619 (1998) · Zbl 0905.35005 · doi:10.1088/0305-4470/31/6/010
[12]Lahno VI, Spichak SV, Stognii VI. Symmetry analysis of evolution type equations. Institute of Computer Science: Moscow-Izhevsk; 2004 [in Russian].
[13]Magadeev BA. On group classification of nonlinear evolution equations. Algebra i Analiz 1993;5:141 – 56 [in Russian]; Translation in St. Petersburg Math J 1994;5:345 – 59. · Zbl 0823.35007
[14]Olver, P.: Applications of Lie groups to differential equations, (1986)
[15]Ono, H.: Algebraic soliton of the modified Korteweg-de Vries equation, J phys soc jpn 41, 1817-1818 (1976)
[16]Ovsiannikov, L. V.: Group analysis of differential equations, (1982) · Zbl 0485.58002
[17]Polyanin, A. D.; Zaitsev, V. F.: Handbook of nonlinear partial differential equations, (2004)
[18]Popovych RO. Classification of admissible transformations of differential equations, in collection of works of institute of mathematics (Institute of Mathematics, Kyiv, Ukraine) 2006;3(2):239 – 54. (Available at lt;http://www.imath.kiev.ua/sim;appmath/Collections/collection2006.pdfgt;). · Zbl 1150.35313
[19]Popovych, R. O.; Ivanova, N. M.: Potential equivalence transformations for nonlinear diffusion-convection equations, J phys A 38, 3145-3155 (2005) · Zbl 1126.35340 · doi:10.1088/0305-4470/38/14/006
[20]Popovych, R. O.; Kunzinger, M.; Eshraghi, H.: Admissible point transformations and normalized classes of nonlinear Schrödinger equations, Acta appl math 109, 315-359 (2010) · Zbl 1216.35146 · doi:10.1007/s10440-008-9321-4
[21]Popovych, R. O.; Vaneeva, O. O.: More common errors in finding exact solutions of nonlinear differential equations: part I, Commun nonlinear sci numer simulat 15, 3887-3899 (2010) · Zbl 1222.35009 · doi:10.1016/j.cnsns.2010.01.037
[22]Tang, X. Y.; Zhao, J.; Huang, F.; Lou, S. Y.: Monopole blocking governed by a modified KdV type equation, St appl math 122, 295-304 (2009) · Zbl 1172.35486 · doi:10.1111/j.1467-9590.2009.00434.x
[23]Vaneeva, O. O.; Johnpillai, A. G.; Popovych, R. O.; Sophocleous, C.: Enhanced group analysis and conservation laws of variable coefficient reaction – diffusion equations with power nonlinearities, J math anal appl 330, 1363-1386 (2007) · Zbl 1160.35365 · doi:10.1016/j.jmaa.2006.08.056
[24]Vaneeva, O. O.; Popovych, R. O.; Sophocleous, C.: Enhanced group analysis and exact solutions of variable coefficient semilinear diffusion equations with a power source, Acta appl math 106, 1-46 (2009)