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Approximate similarity reduction for singularly perturbed Boussinesq equation via symmetry perturbation and direct method. (English) Zbl 1152.81493
Summary: We investigate the singularly perturbed Boussinesq equation in terms of the approximate symmetry perturbation method and the approximate direct method. The similarity reduction solutions and similarity reduction equations of different orders display formal coincidence for both methods. Series reduction solutions are consequently derived. For the approximate symmetry perturbation method, similarity reduction equations of the zero order are equivalent to the Painlevé IV, Painlevé I, and Weierstrass elliptic equations. For the approximate direct method, similarity reduction equations of the zero order are equivalent to the Painlevé IV, Painlevé II, Painlevé I, or Weierstrass elliptic equations. The approximate direct method yields more general approximate similarity reductions than the approximate symmetry perturbation method.

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
34C14 Symmetries, invariants of ordinary differential equations
34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
35A30 Geometric theory, characteristics, transformations in context of PDEs
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References:
[1] Lie S., Vorlesungen über Differentialgleichungen mit Bekannten Infinitesimalen Transformationen (1891)
[2] DOI: 10.1007/978-1-4612-6394-4
[3] DOI: 10.1007/978-1-4757-4307-4
[4] Ovisiannikov L. V., Group Analysis of Differential Equations (1982)
[5] DOI: 10.1007/978-1-4612-4350-2
[6] DOI: 10.1007/3-540-12730-5_12
[7] Bluman G. W., J. Math. Mech. 18 pp 1025– (1969)
[8] DOI: 10.1088/0305-4470/22/15/010 · Zbl 0694.35159
[9] DOI: 10.1063/1.528613 · Zbl 0698.35137
[10] DOI: 10.1016/0375-9601(90)90178-Q
[11] DOI: 10.1098/rspa.1994.0035 · Zbl 0814.35003
[12] DOI: 10.1088/0305-4470/25/9/032 · Zbl 0754.35148
[13] DOI: 10.1063/1.530365 · Zbl 0784.35097
[14] DOI: 10.1070/SM1989v064n02ABEH003318 · Zbl 0683.35004
[15] DOI: 10.1070/SM1989v064n02ABEH003318 · Zbl 0683.35004
[16] Baikov V. A., CRC Handbook of Lie Group Analysis of Differential Equations 3, in: Approximate Transformation Groups and Deformations of Symmetry Lie Algebras (1996)
[17] DOI: 10.1088/0305-4470/22/18/007 · Zbl 0711.35086
[18] DOI: 10.1023/B:ACAP.0000018792.87732.25 · Zbl 1066.76052
[19] DOI: 10.1016/j.cam.2005.11.003 · Zbl 1102.65105
[20] Boussinesq J., Compt. Rend. 72 pp 755– (1871)
[21] Boussinesq J., J. Math. Pures Appl. 7 pp 55– (1872)
[22] Whitham G. B., Linear and Nonlinear Waves (1974) · Zbl 0373.76001
[23] Zakharov V. E., Sov. Phys. JETP 38 pp 108– (1974)
[24] Infeld E., Nonlinear Waves, Solitons and Chaos (1990)
[25] DOI: 10.1016/S0377-0427(98)00186-1 · Zbl 0935.65101
[26] Daripa P., Appl. Math. Comput. 101 pp 159– (1999) · Zbl 0937.76050
[27] DOI: 10.1016/j.euromechflu.2006.02.003 · Zbl 1171.76332
[28] DOI: 10.1016/S0096-3003(01)00166-7 · Zbl 1126.76308
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.