×

A simple transformation for nonlinear waves. (English) Zbl 1037.35504

Summary: A transformation method is proposed to establish a relation between linear and nonlinear wave theories. We show that this transformation can be obtained from the sine-Gordon equation. This new method is simpler than the hyperbolic tangent method in solving differential equations and can be used to get more solutions to a wide class of nonlinear wave equations.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35L70 Second-order nonlinear hyperbolic equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Lan, Huibin; Wang, Kelin, J. Phys. A, 23, 3923 (1990) · Zbl 0718.65084
[2] Jeffrey, A.; Mohamad, M. N., Wave Motion, 14, 369 (1990)
[3] Hereman, W.; Banerjee, P.; Korpel, A.; Assanto, G.; Van Immerzele, A.; Meerpoel, A., J. Phys. A, 19, 607 (1986) · Zbl 0621.35080
[4] Ranerjee, R. S., Int. J. Theor. Phys., 32, 879 (1993) · Zbl 0780.35097
[5] Malfliet, W., Am. J. Phys., 60, 650 (1992) · Zbl 1219.35246
[6] Wang, Xin Yi, Chin. Sci. Bull., 36, 1491 (1991)
[7] Yan, Chuntao, The \(sech^2\) solitary wave and its equivalent transformation, (Exploiting symmetry in applied and numerical analysis. Exploiting symmetry in applied and numerical analysis, 22nd AMS-SIAM Summer Seminar, Fort, Collins (1992)), presentation · Zbl 0917.35116
[8] Drazin, P. G.; Johnson, R. S., Solitons: an introduction, Cambridge texts in applied mathematics 3 (1989), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0661.35001
[9] Yan, Chuntao, Chaos, Solitons Fractals, 4, 2103 (1994) · Zbl 0822.35122
[10] Adams, E. P., (Smithsonian Miscellaneous Collections, Vol. 74 (1922), Smithsonian Institution: Smithsonian Institution Washington), No. 1. Smithsonian mathematical formulae and tables of elliptic functions
[11] Gradshteyn, I. S.; Ryzhik, I. M., Table of integrals, series, and products (1980), Academic Press: Academic Press New York · Zbl 0521.33001
[12] Yan, Chuntao, Something hidden in the Fourier series and its partial sum (1993), Preprint · Zbl 0917.35116
[13] Yan, Chuntao, (Ph.D. thesis (1995), Kansas State University)
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.