×

Exact solitary wave solutions of nonlinear wave equations. (English) Zbl 1054.35032

Summary: The hyperbolic function method for nonlinear wave equations is presented. In support of a computer algebra system, many exact solitary wave solutions of a class of nonlinear wave equations are obtained via the method. The method is based on the fact that the solitary wave solutions are essentially of a localized nature. Writing the solitary wave solutions of a nonlinear wave equation as the polynomials of hyperbolic functions, the nonlinear wave equation can be changed into a nonlinear system of algebraic equations. The system can be solved via Wu elimination or Grobner base method. The exact solitary wave solutions of the nonlinear wave equation are obtained including many new exact solitary wave solutions.

MSC:

35L70 Second-order nonlinear hyperbolic equations
35-04 Software, source code, etc. for problems pertaining to partial differential equations
35C05 Solutions to PDEs in closed form
35Q51 Soliton equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ablowitz, M. J.; Carkson, P. A., Nonlinear Evolution and Inverse Scattering., 47-350 (1991), New York: Cambridge University Press, New York · Zbl 0762.35001
[2] Miura, M. R., Backhand Transformation, 4-156 (1978), Berlin: Springer-Verlag, Berlin
[3] Hirota, R., Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons, Phys. Rev. Lett., 27, 1192-1194 (1971) · Zbl 1168.35423
[4] Wang, M. L.; Zhou, Y. B.; Li, Z. B., Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics, Phys. Lett. A, 213, 67-75 (1996) · Zbl 1125.35401
[5] Shang, Y. D., Explicit and exact solutions for a class of nonlinear wave equations, Acta Appl. Math. Sinica (in Chinese), 23, 1, 21-30 (2000) · Zbl 0952.35075
[6] Li, Z. B.; Zhang, S. Q., Exact solitary wave equations for nonlinear wave equations using symbolic computation, Acta Math. Phys. Sinica (in Chinese), 17, 1, 81-89 (1997) · Zbl 0893.65066
[7] Wu, Wenjun, On zeros of algebraic equations: An application of Ritt principle, Kexue Tongbao (Chinese Science Bulletin), 31, 1, 1-5 (1986) · Zbl 0602.14001
[8] Heegard, C.; Little, J.; Saints, K., Systematic encoding via gröbner bases fro a class of algebraic geometric codes, IEEE Trans. Inform. Theory, IT-41, 1752-1761 (1995) · Zbl 0857.94015
[9] Conte, R.; Musette, M., Link between solitary waves and projective Riccati equation, J. Phys. A: Math. Gen., 25, 2609-2612 (1992) · Zbl 0782.35065
[10] Wahlquist, H. D.; Estabrook, F. B., Prolongation structures and nonlinear evolution equations, J. Math. Phys., 16, 1-7 (1975) · Zbl 0298.35012
[11] Whitham, G. B., Linear and Nonlinear Waves, 44-44 (1974), New York: Wiley, New York · Zbl 0373.76001
[12] Constantin, P.; Foias, C.; Nicolaenko, B., Integral Manifolds and Inertial Manifolds for Dissipative Partial Differtial Equations, 111-118 (1981), New York: Springer-Verlag, New York
[13] Chen, S. R.; Chen, X. J., Completeness relation of squared Jost functions to the NLS equation, Acta Phys. Sinica (in Chinese), 48, 5, 882-886 (1999)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.