Qualitative Analysis and Analytical Wave Solutions for a Particular Scale-Invariant Analog of the Korteweg-de Vries Equation
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Mohamed Atta [1] ; Lewa Alzaleq [1]
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[1] Al al-Bayt University
Al al-Bayt University
Jordania
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- Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 24, Nº 5, 2025
- Idioma: inglés
- Enlaces
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Resumen
The Korteweg–de Vries (KdV) equation has long served as a foundational model in the study of nonlinear wave phenomena. Over time, researchers have developed extensions of the KdV equation to account for more complex physical settings, such as variable media, scaling effects, or external forces. One such extension is the scaleinvariant analog of the KdV (SIdV) equation, which retains some key features of the classical KdV equation–including the sech2 soliton solution–while exhibiting distinct conservation laws and nonlinear dynamics. In this study, we analyze a specific SIdV equation corresponding to the scaling parameter δ = 1 5 , representing a particular case within the broader KdV–SIdV family. This special value of δ is selected because it yields a mathematically tractable structure, enabling both a complete qualitative analysis (e.g., via Hamiltonian phase portraits) and the construction of exact traveling wave solutions—capabilities generally not possible for arbitrary non-zero δ. The analysis includes the derivation of conservation laws, a qualitative investigation through phase portraits, and a detailed classification of solution behaviors. We uncover a range of solution types, including smooth solitary waves, peakon solutions, singular waves, and periodic singular waves. Furthermore, we apply two direct analytical techniques–the tanh-method and the coth-method–to obtain both bounded and unbounded traveling wave solutions. These methods yield novel exact solutions that are not attainable through qualitative phase plane analysis alone. The results presented in this work contribute new findings to the theory of nonlinear dispersive equations and enhance the understanding of the rich solution structure of the SIdV family.
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Referencias bibliográficas
- 1. Hirota, R.: Exact solution of theorteweg-de Vries equation for multiple collisions of solitons. Phys. Rev. Lett. 27(18), 1192 (1971)
- 2. Ablowitz, M.J., Clarkson, P.A.: Solitons, nonlinear evolution equations and inverse scattering (Vol. 149). Cambridge University Press (1991)
- 3. Crighton, D.G.: Applications of KdV. Acta Appl. Math. 39, 39–67 (1995)
- 4. Berest, Y.Y., Loutsenko, I.M.: Huygens’ principle in Minkowski spaces and soliton solutions of the Korteweg-de Vries equation. Commun....
- 5. Hershkowitz, N., Romesser, T.: Observations of ion-acoustic cylindrical solitons. Phys. Rev. Lett. 32(11), 581 (1974)
- 6. Gardner, C.S., Greene, J.M., Kruskal, M.D., Miura, R.M.: Method for solving the Korteweg-deVries equation. Phys. Rev. Lett. 19(19), 1095...
- 7. Sen, A., Ahalpara, D.P., Thyagaraja, A., Krishnaswami, G.S.: A KdV-like advection-dispersion equation with some remarkable properties....
- 8. Alzaleq, L., Manoranjan, V.: Exploring Third-Order KdV-SIdV Families: Analytical Solutions, Conservation Properties, and Phase Plane Trajectories....
- 9. Fan, X., Yin, J.: Two types of traveling wave solutions of a KdV-like advection-dispersion equation. Math. Aeterna 2, 273–282 (2012)
- 10. Wazwaz, A.M.: Peakon and solitonic solutions for KdV-like equations. Phys. Scr. 90(4), 045203 (2015)
- 11. Wang, Z., Liu, X.: Bifurcations and exact traveling wave solutions for the KdV-like equation. Nonlinear Dyn. 95, 465–477 (2019)
- 12. Alzaleq, L.,Manoranjan, V., Alzalg, B.: Exact traveling waves of a generalized Scale-Invariant analogue of the Korteweg-de Vries equation....
- 13. Seadawy, A.R., Ali, A.: Solitary wave solutions of a generalized Scale-Invariant analog of the Kortewegde Vries equation via applications...
- 14. González-Gaxiola, O., Ruiz de Chávez, J.: Traveling wave solutions of the generalized Scale-Invariant analog of the KdV equation by tanh-coth...
- 15. Saifullah, S., Alqarni, M.M., Ahmad, S., Baleanu, D., Khan, M.A., Mahmoud, E.E.: Some more bounded and singular pulses of a generalized...
- 16. Ahmad, S., Aldosary, S.F., Khan, M.A.: Stochastic solitons of a short-wave intermediate dispersive variable (SIdV) equation. AIMS Math....
- 17. da Silva, P.L., Freire, I.L., Sampaio, J.C.S.: A family of wave equations with some remarkable properties. Proceedings of the Royal Society...
- 18. Zhang, G., He, J., Wang, L., Mihalache, D.: Kink-type solutions of the SIdV equation and their properties. Royal Society Open Science...
- 19. Ahalpara, D.P., Sen, A.: A sniffer technique for an efficient deduction of model dynamical equations using genetic programming. In European...
- 20. Kumar, S., Dhiman, S.K.: Exploring cone-shaped solitons, breather, and lump-forms solutions using the lie symmetry method and unified...
- 21. Kumar, S., Rani, S., Mann, N.: Analytical soliton solutions to a (2+1)-dimensional variable coefficients graphene sheets equation...
- 22. Mohan, B., Kumar, S., Kumar, R.: On investigation of kink-solitons and rogue waves to a new integrable (3+1)-dimensional KdV-type...
- 23. Mohan, B., Kumar, S.: Rogue-wave structures for a generalized (3+1)-dimensional nonlinear wave equation in liquid with gas bubbles....
- 24. Kumar, S., Mohan, B.: Bilinearization and new center-controlled N-rogue solutions to a (3+1)- dimensional generalized KdV-type equation...
- 25. Mann, N., Kumar, S.: In-depth analysis and exploration of rogue wave, lump wave, and different solitonic patterns to the Painlevé-integrable...
- 26. Kumar, S., Kukkar, A.: Dynamics of several optical soliton solutions of a (3+1)-dimensional nonlinear Schrödinger equation with parabolic...
- 27. Alzaleq, L., Manoranjan, V.: Qualitative analysis and exact traveling wave solutions for the KleinGordon equation with quintic nonlinearity....
- 28. Al-zaleq, D.A., Alzaleq, L.: Novel soliton solutions and phase plane analysis in nonlinear Schrödinger equations with logarithmic nonlinearities....
- 29. Alzaleq, L., Manoranjan, V.: Analysis of a Reaction-Diffusion-Advection Model with Various Allee Effects. Mathematics 11(10), 2373 (2023)
- 30. Gökta¸s, Ü., Hereman, W.: Symbolic computation of conserved densities for systems of nonlinear evolution equations. J. Symb. Comput. 24(5),...
- 31. Alzaleq, L.: Jacobi elliptic function solutions for the resonant nonlinear Schrödinger equation with anti-cubic nonlinearity. Optik 291,...
- 32. Alzaleq, L., Alzaalig, A., Manoranjan, V.: Exact traveling waves for a generalized Fisher’s equation. Journal of Interdisciplinary Mathematics...
- 33. Nadeem, M., Li, Z., Kumar, D., Alsayaad, Y.: A robust approach for computing solutions of fractionalorder two-dimensional Helmholtz equation....
- 34. Nadeem, M., Li, Z.: Yang transform for the homotopy perturbation method: promise for fractalfractional models. Fractals 31(07), 2350068...
- 35. Nadeem, M., Li, Z.: A new strategy for the approximate solution of fourth-order parabolic partial differential equations with fractional...
- 36. Muhammad, J., Riaz, M.B., Younas, U., Nasreen, N., Jhangeer, A., Lu, D.: Extraction of optical wave structures to the coupled fractional...
- 37. Jhangeer, A., Ehsan, H., Riaz, M.B., Talafha, A.M.: Impact of fractional and integer order derivatives on the (4+1)-dimensional fractional...
- 38. Arif, F., Jhangeer, A., Mahomed, F.M., Zaman, F.D.: Lie group classification and conservation laws of a (2+1)-dimensional nonlinear...
- 39. Kousar, M., Jhangeer, A., Muddassar, M.: Comprehensive analysis of noise behavior influenced by random effects in stochastic differential...
- 40. Infal, B., Jhangeer, A., Muddassar, M.: Dynamical patterns in stochastic ρ4 equation: An analysis of quasi-periodic, bifurcation, chaotic...
- 41. Jhangeer, A., Tariq, K.U., Ali, M.N.: On some new travelling wave solutions and dynamical properties of the generalized Zakharov system....
- 42. Ali, F., Jhangeer, A., Mudassar, M.: A complete dynamical analysis of discrete electric lattice coupled with modified Zakharov-Kuznetsov...
