Abstract
This work researches the singular traveling wave system of the (\(\hbox {n}+1\))-dimensional nonlinear Klein-Gordon equation via the bifurcation theory of dynamical systems. The bifurcations and phase portraits of the traveling wave system are investigated and the influence of singularity and nonlinearity on the dynamical behavior of traveling wave solutions is discussed. Accordingly the various sufficient conditions for the existence of analytic and nonanalytic traveling wave solutions are obtained. Furthermore some exact solutions are given to illustrate the results.
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References
Ablowitz, M., Clarkson, P.: Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge (1991)
Whitham, G.B.: Linear and Nonlinear Waves. Wiley, New York (1974)
Rashidinia, J., Ghasemi, M., Jalilian, R.: Numerical solution of the nonlinear Klein-Gordon equation. J. Comput. Appl. Math. 233, 1866–1878 (2010)
Wazwaz, A.: Solutions of compact and noncompact structures for nonlinear Klein-Gordon-type equation. Appl. Math. Comput. 134, 487–500 (2003)
Tian, L.X., Yu, S.: Nonsymmetrical compacton and multi-compacton of nonlinear intensity KleinCGordon equation. Chaos Solitons Fract. 29, 282–293 (2006)
Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer, New York (1983)
Luo, D., Wang, X., Zhu, D., Han, M.: Bifurcation Theory and Methods of Dynamical Systems. World Scientific, Singapore (1997)
Lawrence, P.: Differential Equations and Dynamical Systems. Springer, New York (1991)
Ma, Z., Zhou, Y., Li, C.: Qualitative and Stability Methods for Ordinary Differential Equations. Science Press, Beijing (2016)
Han, M., Gu, S.: Theory and Method of Nonlinear Systems. Science Press, Beijing (2001)
Rosenau, P., Hyman, J.M.: Compactons: solitons with finite wavelengths. Phys. Rev. Lett. 70, 564–567 (1993)
Li, Y., Olver, P.J., Rosenau, P.: Non-analytic solutions of nonlinear wave models. In: Grosser, M., Hormann, G., Kunzinger, M., Oberguggenberger, M. (eds.) Nonlinear Theory of Generalized Functions, pp. 129–145. Chapman and Hall/CRC, New York (1998)
Li, J.B., Liu, Z.R.: Smooth and non-smooth traveling waves in a nonlinearly dispersive equation. Appl. Math. Model. 25, 41–56 (2000)
Li, J.B., Liu, Z.R.: Travelling wave solutions for a class of nonlinear dispersive equations. Chinese Ann. Math. B 23, 397–418 (2003)
Feng, D.H., Li, J.B.: Dynamics and bifurcations of travelling wave solutions of R(m,n) equations. Proc. Indian Acad. Sci. (Math. Sci.) 117, 555–574 (2007)
Feng, D.H., Lü, J.L., Li, J.B., He, T.L.: Bifurcation studies on traveling wave solutions for nonlinear intensity Klein-Gordon equation. Appl. Math. Comput. 189, 271–284 (2007)
Byrd, P., Friedman, M.: Handbook of Elliptic Integrals for Engineers and Physicists. Spring, Berlin (1954)
Acknowledgements
The authors sincerely thank the reviewers and editors for their valuable comments and suggestions. This research is supported by National Natural Science Foundation of China (Nos. 11662001, 11461021, 11571318, 11761019), Guangxi Natural Science Foundation (No. 2015GXNSFBA139004) and the Training Program for High Level Innovative Talents of Guizhou Province (No. QKH[2017]5658).
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Feng, D., Li, J. & Jiao, J. Dynamical Behavior of Singular Traveling Waves of (\(\hbox {n}+1\))-Dimensional Nonlinear Klein-Gordon Equation. Qual. Theory Dyn. Syst. 18, 265–287 (2019). https://doi.org/10.1007/s12346-018-0285-0
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DOI: https://doi.org/10.1007/s12346-018-0285-0