Yan, Zhenya Generalized transformations and abundant new families of exact solutions for (2 + 1)-dimensional dispersive long wave equations. (English) Zbl 1064.35030 Comput. Math. Appl. 46, No. 8-9, 1363-1372 (2003). The author derives many types of exact solutions of the (2+1)-dimensional dispersive long wave equation by a generalized transformation. In addition the author also obtains many corresponding solitary and periodic wave solutions for the variant Boussinesq equation, as a spherical case of the (2+1)-dimensional dispersive long wave equation. The author shows that, with the aid of symbolic computation and the Wu elimination method, solving nonlinear partial differential equations can be carried out on a computer. Reviewer: Messoud A. Efendiev (Berlin) Cited in 4 Documents MSC: 35C05 Solutions to PDEs in closed form 35G20 Nonlinear higher-order PDEs 35B10 Periodic solutions to PDEs 35Q51 Soliton equations 35A22 Transform methods (e.g., integral transforms) applied to PDEs 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) Keywords:variant Boussinesq equation; Riccati equation; solitary wave solution; periodic wave solution. PDFBibTeX XMLCite \textit{Z. Yan}, Comput. Math. Appl. 46, No. 8--9, 1363--1372 (2003; Zbl 1064.35030) Full Text: DOI References: [1] Ablowitz, M. J.; Clarkson, P. A., Soliton, Nonlinear Evolution Equations and Inverse Scatting (1991), Cambridge University Press: Cambridge University Press New York · Zbl 0762.35001 [2] Gu, C. H., Soliton Theory and Its Application (1990), Zhejiang Science and Technology Press: Zhejiang Science and Technology Press Zhejiang [3] Cox, D., Ideal, Varieties and Algorithms (1991), Springer-Verlag: Springer-Verlag New York [4] Bluman, G. W.; Kumei, S., Symmetry and Differential Equations (1989), Springer-Verlag: Springer-Verlag Berlin · Zbl 0718.35003 [5] Yan, C. T., A simple transformation for nonlinear waves, Phys. Lett. A, 224, 77-82 (1996) [6] Wang, M. L., Applications of a homogeneous balance method to exact solution of nonlinear equations in mathematical physics, Phys. Lett. A, 216, 67-75 (1996) · Zbl 1125.35401 [7] Wang, M. L., Solitary wave solutions for the variant Boussinesq equations, Phys. Lett. A, 199, 169-172 (1995) · Zbl 1020.35528 [8] Yan, Z. Y.; Zhang, H. Q., New explicit and exact travelling wave solutions for a system of variant Boussinesq equations in mathematical physics, Phys. Lett. A, 252, 291-296 (1999) · Zbl 0938.35130 [9] Yan, Z. Y.; Zhang, H. Q., On a new algorithm of constructing solitary wave solutions for systems of nonlinear evolution equations in mathematical physics, Appl. Math. Mech., 21, 382-389 (2000) [10] Ma, W. X., Explicit and exact solutions to a KPP equation, Int. J. Non-Linear Mechanics, 31, 329-338 (1996) · Zbl 0863.35106 [11] Wu, W., On zeros of algebraic equations, Kexue Tongbao, 31, 1-5 (1986) · Zbl 0602.14001 [12] Yan, Z. Y.; Zhang, H. Q., New explicit solitary wave solutions and periodic wave solutions for the WBK equation in shallow water, Phys. Lett. A, 285, 355-362 (2001) · Zbl 0969.76518 [13] Wu, W., Polynomial equations-Solving and its application, (Du, D.; etal., Algorithms and Computation (1994), Springer-Verlag: Springer-Verlag Berlin), 1-6 [14] Yan, Z. Y.; Zhang, H. Q., Explicit exact solutions for the generalized combined KdV-mKdV equation, Appl. Math. J. Chin. Univ. Ser. B, 16, 156-160 (2001) · Zbl 0985.35080 [15] Yan, Z. Y.; Zhang, H. Q., Study on exact analytical solutions for two systems of nonlinear evolution equations, Appl. Math. Mech., 22, 925-933 (2001) 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.