Abstract:
Nonlinear generalizations of integrable equations in one dimension, such as the Korteweg–de Vries and Boussinesq equations with p-power nonlinearities, arise in many physical applications and are interesting from the analytic standpoint because of their critical behavior. We study analogous nonlinear p-power generalizations of the integrable Kadomtsev–Petviashvili and Boussinesq equations in two dimensions. For all p≠0, we present a Hamiltonian formulation of these two generalized equations. We derive all Lie symmetries including those that exist for special powers p≠0. We use Noether's theorem to obtain conservation laws arising from the variational Lie symmetries. Finally, we obtain explicit line soliton solutions for all powers p>0 and discuss some of their properties.
Keywords:
line soliton, conservation law, Kadomtsev–Petviashvili equation.
Citation:
S. C. Anco, M. L. Gandarias, E. Recio, “Conservation laws, symmetries, and line soliton solutions of generalized KP and Boussinesq equations with p-power nonlinearities in two dimensions”, TMF, 197:1 (2018), 3–23; Theoret. and Math. Phys., 197:1 (2018), 1393–1411
\Bibitem{AncGanRec18}
\by S.~C.~Anco, M.~L.~Gandarias, E.~Recio
\paper Conservation laws, symmetries, and line soliton solutions of generalized KP and Boussinesq equations with $p$-power nonlinearities in two dimensions
\jour TMF
\yr 2018
\vol 197
\issue 1
\pages 3--23
\mathnet{http://mi.mathnet.ru/tmf9483}
\crossref{https://doi.org/10.4213/tmf9483}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=3859413}
\adsnasa{https://adsabs.harvard.edu/cgi-bin/bib_query?2018TMP...197.1393A}
\elib{https://elibrary.ru/item.asp?id=35601313}
\transl
\jour Theoret. and Math. Phys.
\yr 2018
\vol 197
\issue 1
\pages 1393--1411
\crossref{https://doi.org/10.1134/S004057791810001X}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000449768100001}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85056081578}
Linking options:
https://www.mathnet.ru/eng/tmf9483
https://doi.org/10.4213/tmf9483
https://www.mathnet.ru/eng/tmf/v197/i1/p3
This publication is cited in the following 24 articles:
Almudena P. Márquez, Tamara M. Garrido, Chaudry Masood Khalique, María L. Gandarias, “Symmetries, conservation laws, and line soliton solutions of a two-dimensional generalized KdV equation with $ p $-power”, DCDS-S, 2024
N. Serikbayev, A. Saparbekova, “Symmetry and conservation laws of the (2+1)-dimensional nonlinear Schrödinger-type equation”, Int. J. Geom. Methods Mod. Phys., 20:10 (2023), 2350172
J. A. Alvarez-Valdez, G. Fernandez-Anaya, “Roadmap of the multiplier method for partial differential equations”, Mathematics, 11:22 (2023), 4572
M. Jafari, S. Mahdion, A. Akgül, S. M. Eldin, “New conservation laws of the Boussinesq and generalized Kadomtsev–Petviashvili equations via homotopy operator”, Results in Physics, 47 (2023), 106369
L. Ju, J. Zhou, Y. Zhang, “Conservation laws analysis of nonlinear partial differential equations and their linear soliton solutions and Hamiltonian structures”, Communications in Analysis and Mechanics, 15:2 (2023), 24
M. Rosa, M.L. Gandarias, A. Niño-López, S. Chulián, “Exact solutions through symmetry reductions for a high-grade brain tumor model with response to hypoxia”, Chaos, Solitons & Fractals, 171 (2023), 113468
S. J. Ryskamp, M. A. Hoefer, G. Biondini, “Modulation theory for soliton resonance and Mach reflection”, Proc. R. Soc. A, 478:2259 (2022)
A. Iqbal, I. Naeem, “Conservation laws and exact solutions of a generalized Kadomtsev–Petviashvili (KP)-like equation”, Math. Methods in App. Sciences, 45:17 (2022), 11206
S. C. Anco, E. Recio, “Topological charges and conservation laws involving an arbitrary function of time for dynamical PDEs”, Proc. R. Soc. A-Math. Phys. Eng. Sci., 477:2245 (2021), 20200442
S. C. Anco, M. L. Gandarias, E. Recio, “Line-solitons, line-shocks, and conservation laws of a universal kp-like equation in 2+1 dimensions”, J. Math. Anal. Appl., 504:1 (2021), 125319
G. R. Deffo, S. B. Yamgoue, T. F. Fozin, F. B. Pelap, “Bifurcation of gap solitary waves in a two-dimensional electrical network with nonlinear dispersion”, Chaos Solitons Fractals, 144 (2021), 110630
F. B. Pelap, J. E. Ndecfo, G. R. Deffo, “Hybrid behavior of a two-dimensional noguchi nonlinear electrical network”, Phys. Scr., 96:7 (2021), 075211
S. C. Anco, B. Wang, “A formula for symmetry recursion operators from non-variational symmetries of partial differential equations”, Lett. Math. Phys., 111:3 (2021), 70
S. A. T. Fonkoua, F. B. Pelap, G. R. Deffo, A. Fomethe, “Rogue wave signals in a coupled anharmonic network: effects of the transverse direction”, Eur. Phys. J. Plus, 136:4 (2021), 416
Acharya S.P., Mukherjee A., Janaki M.S., “Accelerated Magnetosonic Lump Wave Solutions By Orbiting Charged Space Debris”, Nonlinear Dyn., 105:1 (2021), 671–689
Feng Zhang, Yuru Hu, Xiangpeng Xin, Stephen C. Anco, “Lie Symmetry Analysis, Exact Solutions, and Conservation Laws of Variable-Coefficients Boiti-Leon-Pempinelli Equation”, Advances in Mathematical Physics, 2021 (2021), 1
de la Rosa R., Recio E., Garrido T.M., Bruzon M.S., “Lie Symmetry Analysis of (2+1)-Dimensional Kdv Equations With Variable Coefficients”, Int. J. Comput. Math., 97:1-2 (2020), 330–340
S. C. Anco, M. L. Gandarias, “Symmetry multi-reduction method for partial differential equations with conservation laws”, Commun. Nonlinear Sci. Numer. Simul., 91 (2020), 105349
S. C. Anco, M. Luz Gandarias, E. Recio, “Conservation laws and line soliton solutions of a family of modified kp equations”, Discret. Contin. Dyn. Syst.-Ser. S, 13:10, SI (2020), 2655–2665
M. Luz Gandarias, M. Rosa Duran, Ch. Masood Khalique, “Conservation laws and travelling wave solutions for double dispersion equations in (1+1) and (2+1) dimensions”, Symmetry-Basel, 12:6 (2020), 950