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Nonclassical symmetry reductions and exact solutions of the Zabolotskaya–Khokhlov equation

Published online by Cambridge University Press:  16 July 2009

Peter A. Clarkson  and
Simon Hood
Peter A. Clarkson
Affiliation:
Department of Mathematics, University of Exeter, Exeter EX4 4QE, UK
Simon Hood
Affiliation:
Department of Mathematics, University of Exeter, Exeter EX4 4QE, UK
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Abstract

In this paper, new non-classical symmetry reductions and exact solutions for the 2+1- dimensional, time-independent and time-dependent, dissipative Zabolotskaya-Khokhlov equations in both cartesian and cylindrical coordinates, are presented. These are obtained using the Direct Method, which was originally developed by Clarkson & Kruskal (1989) to study symmetry reductions of the Boussinesq equation, and which involves no group theoretic techniques. In particular, we derive exact solutions of these Zabolotskaya-Khokhlov equations expressible in terms of elementary functions, Weierstrass elliptic and zeta functions, Weber-Hermite functions and Airy functions. Additionally, it is shown that some previously known solutions of these equations actually arise from non-classical symmetries.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 3 , Issue 4 , December 1992 , pp. 381 - 414
Copyright
Copyright © Cambridge University Press 1992

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References

Ablowitz, M. J., Ramani, A. & Segur, H. 1980 A connection between nonlinear evolution equations and ordinary differential equations of P-type. I. J. Math. Phys. 21, 715–721.CrossRefGoogle Scholar
Ames, W. F. 1972 Nonlinear Partial Differential Equations in Engineering, II. Academic Press, New York.Google Scholar
Barrera, P. & Brugarino, T. 1986 Similarity solutions of the generalized Kadomtsev-Petviashvili equation. Nuovo Cim. 2 B, 142–156.CrossRefGoogle Scholar
Bartucelli, M., Pantano, P. & Brugarino, T. 1983 Two-dimensional Burgers equation. Lett. Nuovo Cim. 37, 433–438.CrossRefGoogle Scholar
Bluman, G. W. & Cole, J. D. 1969 The general similarity of the heat equation. J. Math. Mech. 18, 1025–1042.Google Scholar
Bluman, G. W. & Cole, J. D. 1974 Similarity methods for differential equations. Appl. Math. Sci. 13, Springer-Verlag, Berlin.Google Scholar
Bluman, G. W. & Kumei, S. 1989 Symmetries and differential equations. Appl. Math. Sci. 81, Springer-Verlag, Berlin.Google Scholar
Buchholz, H. 1969 The Confluent Hypergeometric Equation. Springer-Verlag, Berlin.CrossRefGoogle Scholar
Cates, A. T. 1990 Diffracting nonlinear acoustic beams in 3+1-dimensions with applications to oceanic acoustics. Physica, 44D, 303–312.Google Scholar
Cates, A. T. & Crighton, D. G. 1990 Nonlinear diffraction and caustic formation. Proc. R. Soc. Lond. A 430, 69–88.Google Scholar
Champagne, B., Hereman, W. & Winternitz, P. 1991 The computer calculation of Lie point symmetries of large systems of differential equations. Comp. Phys. Comm. 66, 319–340.CrossRefGoogle Scholar
Champagne, B. & Winternitz, P. 1985 A MACSYMA program for calculating the symmetry group of a system of differential equations, preprint CRM-1278, Montreal.Google Scholar
Clarkson, P. A. 1989a New similarity solutions for the modified Boussinesq equation. J. Phys. A: Math. Gen. 22, 2355–2367.CrossRefGoogle Scholar
Clarkson, P. A. 1989b New similarity solutions and Painlevé analysis for the Symmetric Regularized Long Wave and the Modified Benjamin-Bona-Mahoney equations. J. Phys. A: Math. Gen. 22, 3821–3848.CrossRefGoogle Scholar
Clarkson, P. A. 1990 New exact solutions for the Boussinesq equation. Europ. J. Appl. Math. 1, 279–300.CrossRefGoogle Scholar
Clarkson, P. A. 1992 Dimensional reductions and exact solutions of a generalized nonlinear Schrödinger equation. Nonlinearity 5, 453–472.CrossRefGoogle Scholar
Clarkson, P. A. & Kruskal, M. D. 1989 New similarity solutions of the Boussinesq equation. J. Math. Phys. 30, 2201–2213.CrossRefGoogle Scholar
Clarkson, P. A. & Winternitz, P. 1991 Nonclassical symmetry reductions for the Kadomtsev-Petviashvili equation. Physica 49D, 257–272.Google Scholar
Cole, J. D. 1951 On a quasi-linear parabolic equation occuring in aerodynamics. Q. Appl. Math. 9, 225–236.CrossRefGoogle Scholar
Crighton, D. G. 1979 Model equations of nonlinear acoustics. Ann. Fluid Mech. 11, 11–33.CrossRefGoogle Scholar
Crighton, D. G. 1986 Basic theoretical nonlinear acoustics, Frontiers in Physical Acoustics (ed. Sette, D.), North Holland, pp. 1–50.Google Scholar
Crighton, D. G. & Scott, J. F. 1979 Asymptotic solutions of model equations in nonlinear acoustics. Phil. Trans. R. Soc. Lond. A 292, 107–134.Google Scholar
Gaeta, G. 1990 On the conditional symmetries of Levi and Winternitz. J. Phys. A: Math. Gen. 23, 3643–3645.CrossRefGoogle Scholar
Gibbons, J. & Kodama, Y. 1988 Integrable quasi-linear systems: generalized hodograph transformation, Nonlinear Evolutions (ed. Leon, J. J. P.), World Scientific, Singapore, pp. 97–107.Google Scholar
Hansen, A. G. 1964 Similarity Analyses of Boundary Value Problems in Engineering, Prentice-Hall, Englewood Cliffs.Google Scholar
Head, A. 1990 Lie: a MUMATH program for the calculation of the Lie algebra of differential equations, preprint.Google Scholar
Hill, J. M 1982 Solution of Differential Equations by means of One-parameter Groups, Research Notes Math., 63, Pitman, Boston.Google Scholar
Hirota, R. 1980 Direct methods in soliton theory, Solitons, (eds. Bullough, R. K. and Caudrey, P. J.), Topics in Current Physics, 17, Springer-Verlag, Berlin, pp. 157–176.CrossRefGoogle Scholar
Hopf, E. 1950 The partial differential equation ut+uux = μuxx. Commun. Pure Appl. Math. 3, 201–230.CrossRefGoogle Scholar
Hunter, J. K. 1988 Transverse diffraction of nonlinear waves and singular rays. SIAM J. Appl. Math. 48, 1–37.CrossRefGoogle Scholar
Ince, E. L. 1956 Ordinary Dfferenial Equations. Dover, New York.Google Scholar
Kersten, P. H. M. 1987 Infinitesimal Symmetries: a Computational Approach, CWI Tract, 34, Amsterdam.Google Scholar
King, J. R. 1991 Exact results for the nonlinear diffusion equation ∂u/∂t = (∂/∂x)(u-4/3∂u/∂x) and ∂u/∂t = (∂/∂x) (u-2/3∂u/∂x). J. Phys. A: Math. Gen. 24, 5721–5745.CrossRefGoogle Scholar
Kodama, Y. 1988a A method for solving the dispersionless KP equation and its exact solutions. Phys. Lett. 129A, 223–226.CrossRefGoogle Scholar
Kodama, Y. 1988b A solution method for the dispersionless KP equations and its exact solutions. Prog. Theo. Phys. Supp. 94, 184–194.CrossRefGoogle Scholar
Kuznetsov, V. P. 1970 Equations of nonlinear acoustics. Soviet Phys. Acoustics 16, 467–470.Google Scholar
Levi, D. & Winternitz, P. 1989 Nonclassical symmetry reduction: example of the Boussinesq equation. J. Phys. A: Math. Gen. 22, 2915–2924.CrossRefGoogle Scholar
Lou, S.-Y. 1990a Similarity solutions of the Kadomtsev-Petviashvili equation. J. Phys. A: Math. Gen. 23, L649–654.CrossRefGoogle Scholar
Lou, S.-Y. 1990b A note on the new similarity reductions of the Boussinesq equation. Phys. Lett. 151A, 133–135.CrossRefGoogle Scholar
Lou, S.-Y. 1991 Generalized Boussinesq equation and KdV equation-Painlevé properties, Backlund transformations and Lax pairs. Sci. China, Ser. A 34, 1098–1108.Google Scholar
Lou, S.-Y. & Ni, G.-J. 1991 Some new solutions of the KP equation. Commun. Theor. Phys. 15, 465–472.CrossRefGoogle Scholar
Lou, S.-Y., Ruan, H.-Y., Chen, D.-F. & Chen, W.-Z. 1991 Similarity reductions of the KP equation by a direct method. J. Phys. A: Math. Gen. 24, 1455–1467.CrossRefGoogle Scholar
Nucci, M. C. 1990 Preprint Math:062090–051, School of Mathematics, Georgia Institute of Technology, Atlanta.Google Scholar
Nucci, M. C. & Clarkson, P. A. 1992 The nonclassical method is more general than the direct method for symmetry reductions: an example of the Fitzhugh-Nagumo equation. Phys. Lett. 164A, 49–56.CrossRefGoogle Scholar
Oliveri, F. 1991 Painlevé analysis and similarity solutions of Burgers' equation with variable coefficients. J. Eng. Math. 25, 317–327.CrossRefGoogle Scholar
Olver, P. J. 1986 Applications of Lie Groups to Differential Equations, Graduate Texts Math. 107, Springer-Verlag, New York.CrossRefGoogle Scholar
Olver, P. J. & Rosenau, P. 1986 The construction of special solutions to partial differential equations. Phys. Lett. 114A, 107–112.CrossRefGoogle Scholar
Olver, P. J. & Rosenau, P. 1987 Group-invariant solutions of differential equations. SIAMJ. Appl. Math. 47, 263–275.CrossRefGoogle Scholar
Oron, A. & Rosenau, P. 1986 Some symmetries of the nonlinear heat and wave equations. Phys. Lett. 118A, 172–176.CrossRefGoogle Scholar
Pucci, E. & Saccomandi, G. 1992 On the weak symmetry groups of partial differential equations. J. Math. Anal. Appl. 163, 588–598.CrossRefGoogle Scholar
Quispel, G. R. W. & Capel, H. W. 1983 The anisotropic Heisenberg spin chain and the nonlinear Schrödinger equation. Physica 117A, 76–102.CrossRefGoogle Scholar
Quispel, G. R. W., Nijhoff, F. W. & Capel, H. W. 1982 Linearization of the Boussinesq equation and the modified Boussinesq equation. Phys. Lett. 91A, 143–145.CrossRefGoogle Scholar
Reid, G. J. 1990a A triangularization algorithm which determines the Lie symmetry algebra of any system of PDEs. J. Phys. A: Math. Gen. 23, L853–859.CrossRefGoogle Scholar
Reid, G. J. 1990b Algorithmic determination of Lie symmetry algebras of differential equations. Lie Theory, Differential Equations and Representation Theory, (ed. Hussin, V.), Les Publications de CRM, Montreal, pp. 363–372.Google Scholar
Reid, G. J. 1991a Algorithms for reducing a system of PDEs to standard form, determining the dimension of its solution space and calculating its Taylor series solution. Europ. J. Appl. Math. 2,293–318.CrossRefGoogle Scholar
Reid, G. J. 1991b Finding abstract Lie symmetry algebras of differential equations without integrating determining equations. Europ. J. Appl. Math. 2, 319–340.CrossRefGoogle Scholar
Rosenau, P. & Schwarzmeier, J. L. 1986 On similarity solutions of Boussinesq type equations. Phys. Lett. 115A, 75–77.CrossRefGoogle Scholar
Schwarz, F. 1985 Automatically determining symmetries of partial differential equations. Computing 34, 91–106.CrossRefGoogle Scholar
Schwarz, F. 1987 Symmetries of the Khokhlov-Zabolotskaya equation. Comment on “Towards the conservation laws and Lie symmetries for the Khokhlov-Zabolotskaya equation in three dimensions. J. Phys. A: Math. Gen. 20, 1613–1614.CrossRefGoogle Scholar
Schwarz, F. 1988 Symmetries of differential equations: from Sophus Lie to computer algebra. SIAM Rev. 30, 450–481.CrossRefGoogle Scholar
Sharomet, N. O. 1989 Symmetries, invariant solutions and conservation laws of the nonlinear acoustic equation. Symmetries of partial differential equations. Acta Appl. Math. 15, 83–120.CrossRefGoogle Scholar
Tajiri, M., Kawamoto, S. & Thushima, K. 1983 Reduction of Burgers' equation to Riccati equation. Math. Japonica 28, 125–133.Google Scholar
Valenti, G. 1990 Transverse diffraction of nonlinear waves in radiative megnetogasdynamics. Wave Motion 12, 97–106.CrossRefGoogle Scholar
Vinogradov, A. M. & Vorob'ev, E. M. 1976 Use of symmetries to find exact solutions of the Zabolotskaya-Khokhlov equation. Soviet Phys. Acoustics 22, 12–15.Google Scholar
Webb, G. M. & Zank, G. P. 1990a Painlevé analysis of the two-dimensional Burgers equation. J. Phys. A: Math. Gen. 23, 5465–5477.CrossRefGoogle Scholar
Webb, G. M. & Zank, G. P. 1990b Painlevé analysis of the three-dimensional Burgers equation. Phys. Lett. 150A, 14–22.CrossRefGoogle Scholar
Weiss, J. 1984 The Sine-Gordon equations: complete and partial integrability. J. Math. Phys. 25, 2226–2235.CrossRefGoogle Scholar
Weiss, J., Tabor, M. & Carnevale, G. 1983 The Painlevé property for partial differential equations. J. Math. Phys. 24, 522–526.CrossRefGoogle Scholar
Whittaker, E. E. & Watson, G. M. 1927 Modern Analysis, 4th ed.Cambridge University Press.Google Scholar
Winternitz, P. 1990 Conditional symmetries and conditional integrability for nonlinear systems, preprint CRM-1709, Montreal.Google Scholar
Zabolotskaya, E. A. & Khokhlov, R. V. 1969 Quasi-plane waves in the nonlinear acoustics of confined beams. Sov. Phys. Acoustics 15, 35–40.Google Scholar
Zabolotskaya, E. A. & Khokhlov, R. V. 1970 Convergent and divergent sound beams in nonlinear media. Sov. Phys. Acoustics 16, 39–43.Google Scholar
Zank, G. P. & Webb, G. M. 1990 Weakly multi-dimensional cosmic-ray-modified MHD shocks. Plasma Phys. 44, 91–101.CrossRefGoogle Scholar