Abstract
This paper considers a generalized Novikov–Veselov equation \(u_{t}+ a u_{xxx}+b u_{yyy}+c (u^{2}\partial _{y}^{-1}u_{x})_{x}+d(u^{2}\partial _{x}^{-1}u_{y})_{y}=0\). By utilizing the bifurcation method and qualitative theory of dynamical systems, we show that there exist two kinds of important bifurcation phenomena when the bifurcation parameters vary. One of the bifurcation phenomena is that the fractional solitary waves can be bifurcated from three types of nonlinear waves which are the trigonometric periodic waves, the elliptic periodic waves and the hyperbolic solitary waves. Another bifurcation phenomenon is that the kink waves can be bifurcated from two types of nonlinear waves which are the solitary waves and the singular waves. Moreover, we carry out many numerical simulations of bifurcation processes of these nonlinear waves to verify our main results.
Similar content being viewed by others
References
Veselov, A.P., Novikov, S.P.: Finite-gap 2-dimensional Schrödinger potential operators. Explicit formula and evolutional equations. Dokl. Akad. Nauk SSSR 279, 20–24 (1984)
Novikov, S.P., Veselov, A.P.: Two-dimensional Schrödinger operator: inverse scattering transform and evolutional equations. Phys. D 18, 267–273 (1986)
Cheng, Y.: Integrable systems associated with the Schrödinger spectral problem in the plane. J. Math. Phys. 32, 157–162 (1991)
Tagami, Y.: Soliton-like solutions to a (2+1)-dimensional generalization of the KdV equation. Phys. Lett. A 141, 116–120 (1989)
Hu, X.B.: Nonlinear superposition formula of the Novikov–Veselov equation. J. Phys. A Math. Gen. 27, 1331–1338 (1994)
Hu, X.B., Willox, R.: Some new exact solutions of the Novikov–Veselov equation. J. Phys. A Math. Gen. 29, 4589–4592 (1996)
Nickel, J., Schürmann, H.W.: 2-Soliton-solution of the Novikov–Veselov equation. Int. J. Theor. Phys. 45, 1825–1829 (2006)
Chang, J.H.: On the N-solitons solutions in the Novikov–Veselov equation. Symmetry Integr. Geom. Methods Appl. 9, 1–13 (2013)
Dubrovsky, V.G., Topovsky, A.V.: About simple nonlinear and linear superpositions of special exact solutions of Veselov–Novikov equation. J. Math. Phys. 54, 1–14 (2013)
Perry, P.A.: Miura maps and inverse scattering for the Novikov–Veselov equation. Anal. PDE 7, 311–343 (2014)
Wazwaz, A.M.: New solitary wave and periodic wave solutions to the (2+1)-dimensional Nizhnik–Novikov–Vesselov system. Appl. Math. Comput. 187, 1584–1591 (2007)
Wazwaz, A.M.: Structures of multiple soliton solutions of the generalized, asymmetric and modified Nizhnik–Novikov–Veselov equations. Appl. Math. Comput. 218, 11344–11349 (2012)
Boubir, B., Triki, H., Wazwaz, A.M.: Bright solitons of the variants of the Novikov–Veselov equation with constant and variable coefficients. Appl. Math. Model. 37, 420–431 (2013)
Li, J.B., Liu, Z.R.: Smooth and non-smooth traveling waves in a nonlinearly dispersive equation. Appl. Math. Model. 25, 41–56 (2000)
Liu, Z.R., Qian, T.F.: Peakons of the Camassa–Holm equation. Appl. Math. Model. 26, 473–480 (2002)
Rui, W.G., He, B., Xie, S.L., Long, Y.: Application of the integral bifurcation method for solving modified Camassa–Holm and Degasperis–Procesi equations. Nonlinear Anal. 71, 3459–3470 (2009)
Song, M., Cao, J., Guan, X.L.: Application of the bifurcation method to the Whitham–Broer–Kaup–Like equations. Math. Comput. Model. 55, 688–696 (2012)
Liu, R., Yan, W.F.: Some common expressions and new bifurcation phenomena for nonlinear waves in a generalized mKdV equation. Int. J. Bifurc. Chaos 23, 1330007 (2013)
Biswas, A., Song, M.: Soliton solution and bifurcation analysis of the Zakharov–Kuznetsov–Benjamin–Bona–Mahoney equation with power law nonlinearity. Commun. Nonlinear Sci. Numer. Simul. 18, 1676–1683 (2013)
Wen, Z.S.: Bifurcation of solitons, peakons, and periodic cusp waves for \(\theta \)-equation. Nonlinear Dyn. 77, 247–253 (2014)
Song, M.: Nonlinear wave solutions and their relations for the modified Benjamin–Bona–Mahony equation. Nonlinear Dyn. 80, 431–446 (2015)
Acknowledgments
This was supported by the National Natural Science Foundation of China (No. 11571116).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhu, N., Pan, C. & Liu, Z. Two kinds of important bifurcation phenomena of nonlinear waves in a generalized Novikov–Veselov equation. Nonlinear Dyn 83, 1311–1324 (2016). https://doi.org/10.1007/s11071-015-2404-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-015-2404-7