In this article, we address the Kadomtsev–Petviashvili (KP) equation in which a small competing dispersion effect is present. We examine the nature of solutions under the influence of dispersion effect by exploiting dynamical system theory and Lyapunov function. We prove the existence of bounded traveling wave solutions when KP equation has external dispersion effect. The retrieved traveling wave solutions are in the form of solitary waves, periodic and elliptic functions. We obtain the general solution of the equation in presence and absence of the dispersion effect in terms of Weirstrass \(\wp \) functions and Jacobi elliptic functions. Furthermore, we apply a new method which is based on factorization method, use of functional transformation and the Abel’s first order nonlinear equation and obtain a new form of kink-type solutions. Finally, we analyze the stability analysis of the dispersive solutions explicitly by using dynamical system theory. We show that the traveling wave velocity behaves like a bifurcation parameter which classifies different classes of waves. Moreover, we also obtain a transcritical bifurcation which occurs at a critical velocity.
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Korteweg, D.J., De Vries, G.: On the Change of Form of Long Waves Advancing in a Rectangular Canal, and on a New Type of Long Stationary Waves, vol. 5, pp. 422–443. Philosophical Magazine, London (1895)
Washimi, H., Taniuti, T.: Propagation of ion-acoustic solitary waves of small amplitude. Phys. Rev. Lett. 17, 996 (1966)
Benney, D.J.: Long non-linear waves in fluid flows. J. Math. Phys. 45, 52 (1966)
Zabusky, N.J., Kruskal, M.D.: Interaction of soliton in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15, 240 (1965)
Fermi, E., Pasta, J., Ulam, S.: Nonlinear Wave Motion, vol. 15. American Mathematical Society, New York (1974)
Thyagaraja, A.: Recurrent motions in certain continuum dynamical systems. Phys. Fluids 22, 2093 (1979)
Thyagaraja, A.: Recurrence phenomena and the number of effective degrees of freedom in nonlinear wave motion, Chap 17. In: Debnath, L. (ed.) Nonlinear Waves, pp. 308–325. Cambridge University Press, Cambridge (1983). https://doi.org/10.1017/CBO9780511569500.018
Kadomtsev, B.B., Petviashvili, V.I.: On the stability of solitary waves in weakly dispersing media. Sov. Phys. Dokl. 15, 539 (1970)
Santini, P.M.: On the evolution of two-dimensional packets of water waves over an uneven bottom. Lett. Nuovo Cim. 30, 236 (1981)
David, D., Levi, D., Winternitz, P.: Integrable nonlinear equations for water waves in straits of varying depth and width. Stud. Appl. Math. 76, 133 (1987a)
David, D., Levi, D., Winternitz, P.: Solitons in shallow seas of variable depth and in marine straits. Stud. Appl. Math. 80, 1 (1989)
Bhrawy, A.H., Abdelkawy, M.A., Kumar, S., Biswas, A.: Solitons and other solutions to Kadomtsev–Petviashvili equation of B-type. Rom. J. Phys. 58, 729 (2013)
Ebadi, G., Fard, N.Y., Bhrawy, A.H., Kumar, S., Triki, H., Yildirim, A., Biswas, A.: Solitons and other solutions to the \((3+1)\)-dimensional extended Kadomtsev-Petviashvili equation with power law nonlinearity. Rom. Rep. Phys 65, 27 (2013)
Jordan, D.W., Smith, P.: Nonlinear Ordinary Differential Equations (An introduction to dynamical systems). Oxford University Press, Oxford (1999)
Alligood, K.T., Sauer, T., Yorke, J.A.: Chaos: An Introduction to Dynamical Systems. Springer-Verlag, New York (1997)
Chow, S.N., Hale, J.K.: Method of Bifurcation Theory. Springer-Verlag, New York (1981)
Ganguly, A., Das, A.: Explicit solutions and stability analysis of the \((2+1)\) dimensional KP-BBM equation with dispersion effect. Commun. Nonlinear Sci. Numer. Simulat. 25, 102–117 (2015)
Das, A.: Explicit Weierstrass traveling wave solutions and bifurcation analysis for dissipative Zakharov-Kuznetsov modified equal width equation. Comput. Appl. Math. (2017). https://doi.org/10.1007/s40314-017-0508-z
Rosu, H.C., Cornejo-Pérez, O.: Supersymmetric pairing of kinks for polynomial nonlinearities. Phys. Rev. E 71, 046607 (2005)
Cornejo-Pérez, O., Rosu, H.C.: Nonlinear second order Ode’s: factorizations and particular solutions. Prog. Theor. Phys. 114, 533 (2005)
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Das, A. Explicit Weierstrass \(\wp \) Traveling Wave Solution and Stability of Dispersive Solutions to the Kadomtsev–Petviashvili Equation with Competing Dispersion Effect. Differ Equ Dyn Syst (2020). https://doi.org/10.1007/s12591-020-00523-x
- KP equation
- Dispersion effect
- Weirstrass \(\wp \) function
- Jacobi elliptic function
- Dynamical system