Abstract
First we perform a symmetry analysis on the KdV equation, and obtain the traveling-wave solutions in terms of the functions ℘(z),tan2 z,coth2 z, and \(\operatorname{cn}^{2}(z\mid m)\), respectively. The two-soliton solution is also determined via the Hirota bilinear presentation. The KP equation is an extension of the KdV equation. We use symmetry transformations to extend the solutions of the KdV equation to the more sophisticated solutions of the KP equation. Then we solve the KP equation for solutions that are polynomial in a spatial variable, and obtain another type of solution. We also find the Hirota bilinear presentation of the KP equation and obtain the “lump” solution. Using the stable range of the nonlinear term, symmetry transformations, and the generalized power series method, we find a family of singular solutions with seven arbitrary parameter functions of t and a family of analytic solutions with six arbitrary parameter functions of t for the equation of transonic gas flows. Similar solutions are also obtained for the short-wave equation and the Khokhlov–Zabolotskaya equation in nonlinear acoustics of bounded bundles. The symmetry transformations and two new families of exact solutions with multiple parameter functions for the geopotential equation are derived.
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References
Ablowitz, M.J., Clarkson, P.A.: Solitons, Nonlinear Evolution Equations and Inverse Scattering. London Math. Soc Lect. Notes, vol. 149. Cambridge University Press, Cambridge (1991)
Bagdoev, A.G., Petrosyan, L.G.: Justification of the applicability of short wave equations in obtaining an equation for modulation of a gas–fluid mixture. Izv. Akad. Nauk Armyan. SSR Ser. Mekh. 38(4), 58–66 (1985)
Cao, B.: Solutions of Jimbo–Miwa equation and Konopelchenko–Dubrovsky equations. Acta Appl. Math. 112, 181–203 (2010)
Ermakov, S.: Short wave/long wave interaction and amplification of decimeter-scale wind waves in film slicks. Geophys. Res. Abstr. 8, 00469 (2006)
Gibbons, J.: The Khokhlov–Zabolotskaya equation and the inverse scattering problem of classical mechanics. In: Dynamical Problems in Soliton Systems, Kyoto, 1984, pp. 36–41. Springer, Berlin (1985)
Hirota, R.: Exact solutions of the Korteweg–de Vries equation for multiple collisions of solitons. Phys. Rev. Lett. 27, 1192–1194 (1971)
Jimbo, M., Miwa, T.: Solitons and infinite dimensional Lie algebras. Publ. RIMS Kyoto Univ. 19, 943–1001 (1983)
Kadomitsev, B.B., Petviashvili, V.I.: On the stability of solitary waves weakly dispersive media. Sov. Phys. Dokl. 15, 539–541 (1970)
Katkov, V.I.: One class of exact solutions of the geopotential forecast equation. Izv. Akad. Nauk SSSR, Fiz. Atmos. Okeana 1, 1088 (1965)
Katkov, V.I.: Exact solutions of the geopotential forecast equation. Izv. Akad. Nauk SSSR, Fiz. Atmos. Okeana 2, 1193 (1966)
Khamitova, R.S.: Group structure and a basis of conservation laws. Teor. Mat. Fiz. 52(2), 244 (1982)
Khokhlov, R.V., Zabolotskaya, E.A.: Quasi-plane waves in nonlinear acoustics of bounded bundles. Akust. Zh. 15(1), 40 (1969)
Khristianovich, S.A., Razhov, O.S.: On nonlinear reflection of weak shock waves. Prikl. Mat. Meh. 22(5), 586 (1958)
Kibel’, T.W.: Introduction to Hydrodynamical Methods of Short Term Weather Forecast. Gosteoretizdat, Moscow (1954)
Kocdryavtsev, A., Sapozhnikov, V.: Symmetries of the generalized Khokhlov–Zabolotskaya equation. Acoust. Phys. 4, 541–546 (1998)
Korsunskii, S.V.: Self-similar solutions of two-dimensional equations of Khokhlov–Zabolotskaya type. Mat. Fiz. Nelinein. Mekh. 16, 81–87 (1991)
Kostin, I., Panasenko, G.: Khokhlov–Zabolotskaya–Kuzentsov-type equation: nonlinear acoustics in heterogeneous media. SIMA J. Math. 40, 699–715 (2008)
Kraenkel, R., Manna, M., Merle, V.: Nonlinear short-wave propagation in ferrites. Phys. Rev. E 61, 976–979 (2000)
Kucharczyk, P.: Group properties of the “short waves” equations in gas dynamics. Bull. Acad. Pol. Sci., Sér. Sci. Tech. XIII(5), 469 (1965)
Kupershmidt, B.A.: Geometric-Hamiltonian Forms for the Kadomtsev and Petviashvili and Khokhlov–Zabolotskaya Equations. In: Geometry in Partial Differential Equations, pp. 155–172. World Scientific, River Edge (1994)
Lin, J., Zhang, J.: Similarity reductions for the Khokhlov–Zabolotskaya equation. Commun. Theor. Phys. 24(1), 69–74 (1995)
Lin, C.C., Reissner, E., Tsien, H.S.: On two-dimensional non-steady motion of a slender body in a compressible fluid. J. Math. Phys. 27(3), 220 (1948)
Mamontov, E.V.: On the theory of nonstationary transonic flows. Dokl. Akad. Nauk SSSR 185(3), 538 (1969)
Mamontov, E.V.: Analytic perturbations in a nonstationary transonic stream. Dinamika Splosh. Sredy 10, 217–222 (1972)
Morozov, O.: Cartan’s structure theory of symmetry pseudo-groups for the Khokhlov–Zabolotskaya equation. Acta Appl. Math. 101, 231–241 (2008)
Roy, C., Nasker, M.: Towards the conservation laws and Lie symmetries for the Khokhlov–Zabolotskaya equation in three dimensions. J. Phys. A 19(10), 1775–1781 (1986)
Roy, S., Roy, C., De, M.: Loop algebra of Lie symmetries for a short-wave equation. Int. J. Theor. Phys. 27(1), 47–55 (1988)
Rozanova, A.: The Khokhlov–Zabolotskaya–Kuznetsov equation. Math. Acad. Sci. Paris 344, 337–342 (2007)
Rozanova, A.: Qualitative analysis of the Khokhlov–Zabolotskaya equation. Math. Models Methods Appl. Sci. 18, 781–812 (2008)
Sanchez, D.: Long waves in ferromagnetic media, Khokhlov–Zabolotskaya equation. J. Differ. Equ. 210, 263–289 (2005)
Schwarz, F.: Symmetries of the Khokhlov–Zabolotskaya equation. J. Phys. A 20(6), 1613 (1987)
Sevost’janov, G.D.: An equation for nonstationary transonic flows of an ideal gas. Izv. Akad. Nauk SSSR Mekh. Zidk. Gaza 1, 105–109 (1977)
Sukhinin, S.V.: Group property and conservation laws of the equation of transonic motion of gas. Dinamika Splosh. Sredi 36, 130 (1978)
Syono, S.: Various simple solutions of the barotropic vorticity equation. In: Vortex, Collected Papers of the Numerical Weather Prediction Group in Tokyo, vol. 3 (1958)
Vinogradov, A.M., Vorob’ev, E.M.: Application of symmetries for finding of exact solutions of Khokhlov–Zabolotskaya equation. Akust. Zh. 22(1), 22 (1976)
Xu, X.: Stable-range approach to the equation of nonstationary transonic gas flows. Q. Appl. Math. 65, 529–547 (2007a)
Xu, X.: Stable-Range approach to short wave and Khokhlov–Zabolotskaya equations. Acta Appl. Math. 106, 433–454 (2009b)
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Xu, X. (2013). Nonlinear Scalar Equations. In: Algebraic Approaches to Partial Differential Equations. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36874-5_5
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DOI: https://doi.org/10.1007/978-3-642-36874-5_5
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