Abstract
In this article, we study a generalized equation that represents a family of compacton-supporting equations. We present the general form of conservation laws for the family of compacton equations by employing the multiplier approach. The generalized conservation laws presented in this study are important in determining the solution process of any third-order nonlinear dispersive partial differential equation that belongs to the category of the generalized compacton family equation. The double reduction theory is employed to construct reductions and new exact solutions of various sub-cases of the compacton family equation.
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References
P. Rosenau, J.M. Hyman, Compactons: solitons with finite wavelength. Phys. Rev. Lett. 70(5), 564–567 (1993)
P. Rosenau, On a model equation of traveling and stationary compactons. Phys. Lett. A 356(1), 44–50 (2006)
P. Rosenau, A. Oron, On compactons induced by a non-convex convection. Commun. Nonlinear Sci. Numer. Simul. 19(5), 1329–1337 (2014)
P. Rosenau, A. Oron, Flatons: flat-top solitons in extended Gardner-like equations. Commun. Nonlinear Sci. Numer. Simul. 91, 105442 (2020)
A. Sergyeyev, S. Skurativskyi, V. Vladimirov, Compacton solutions and (non) integrability of nonlinear evolutionary PDEs associated with a chain of prestressed granules. Nonlinear Anal. Real World Appl. 47, 68–84 (2019)
P. Laplace, Traité de mécanique céleste, vol. 1, paris, 1798, English translation, Celestial Mechanics, New York
E. Noether, Invariant variation problems. Transp. Theory Stat. Phys. 1(3), 186–207 (1971)
A.H. Kara, F.M. Mahomed, Noether-type symmetries and conservation laws via partial Lagrangians. Nonlinear Dyn. 45(3–4), 367–383 (2006)
H. Steudel, Über die zuordnung zwischen lnvarianzeigenschaften und erhaltungssätzen. Z. Naturforschung A 17(2), 129–132 (1962)
P. J. Olver, Applications of Lie Groups to Differential Equations, vol. 107 (Springer, 2000)
R. Naz, F.M. Mahomed, D. Mason, Comparison of different approaches to conservation laws for some partial differential equations in fluid mechanics. Appl. Math. Comput. 205(1), 212–230 (2008)
G. Bluman, S. Anco, Symmetry and Integration Methods for Differential Equations, vol. 154 (Springer, 2008)
S. Kumar, S. Rani, Lie symmetry reductions and dynamics of soliton solutions of \(\left(2 + 1\right)\)-dimensional Pavlov equation. Pramana 94(1), 1–12 (2020)
S. Kumar, S. Rani, Invariance analysis, optimal system, closed-form solutions and dynamical wave structures of a \(\left(2 + 1\right)\)-dimensional dissipative long wave system. Phys. Scr. 96(12), 125202 (2021)
S. Kumar, S. Rani, Lie symmetry analysis, group-invariant solutions and dynamics of solitons to the \(\left(2 + 1\right)\)-dimensional bogoyavlenskii-schieff equation. Pramana 95(2), 1–14 (2021)
S. Rani, S. Kumar, R. Kumar, Invariance analysis for determining the closed-form solutions, optimal system, and various wave profiles for a \(\left(2 + 1\right)\)–dimensional weakly coupled b-type Kadomtsev–Petviashvili equations. J. Ocean Eng. Sci. (2021)
A. Kara, F. Mahomed, Relationship between symmetries and conservation laws. Int. J. Theor. Phys. 39(1), 23–40 (2000)
A. Sjöberg, Double reduction of PDEs from the association of symmetries with conservation laws with applications. Appl. Math. Comput. 184(2), 608–616 (2007)
A. Sjöberg, On double reductions from symmetries and conservation laws. Nonlinear Anal. Real World Appl. 10(6), 3472–3477 (2009)
R. Naz, M. Khan, I. Naeem, Conservation laws and exact solutions of a class of non linear regularized long wave equations via double reduction theory and lie symmetries. Commun. Nonlinear Sci. Numer. Simul. 18(4), 826–834 (2013)
A.H. Bokhari, A.Y. Al-Dweik, A. Kara, F. Mahomed, F. Zaman, Double reduction of a nonlinear (2+ 1) wave equation via conservation laws. Commun. Nonlinear Sci. Numer. Simul. 16(3), 1244–1253 (2011)
A. Zaidi, M. Khan, I. Naeem, Conservation laws and exact solutions of generalized nonlinear system and Nizhink–Novikov–Veselov equation, in Mathematical Problems in Engineering (2018)
A. Iqbal, I. Naeem, Generalised conservation laws, reductions and exact solutions of the \(k \left( m, n \right)\) equations via double reduction theory. Pramana 95(1), 1–9 (2021)
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Iqbal, A., Naeem, I. Conservation laws and exact solutions of a family of compacton-supporting equations. Eur. Phys. J. Plus 137, 535 (2022). https://doi.org/10.1140/epjp/s13360-022-02738-z
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DOI: https://doi.org/10.1140/epjp/s13360-022-02738-z