Ablowitz, Mark J.; Beals, Richard; Tenenblat, Keti On the solution of the generalized wave and generalized sine-Gordon equations. (English) Zbl 0626.35082 Stud. Appl. Math. 74, 177-203 (1986). The direct and inverse scattering problems associated with the n- dimensional generalized wave equation (GWE) and the generalized sine Gordon equation (GSGE) have been solved for initially prescribed data along certain lines. The methodology involves reduction of the n- dimensional equation into a coupled system of n linear one-dimensional ordinary differential equations which could in turn be subject to detailed scattering analysis. The linear scattering problem is more amenable particularly to equations in more than three independent variables for which generally one has to consider scattering data which satisfy nonlinear equations. The present analysis avoids this constraint. Reviewer: P.Chandran Cited in 1 ReviewCited in 26 Documents MSC: 35Q99 Partial differential equations of mathematical physics and other areas of application 35P25 Scattering theory for PDEs 35R30 Inverse problems for PDEs 35L75 Higher-order nonlinear hyperbolic equations 35A30 Geometric theory, characteristics, transformations in context of PDEs Keywords:inverse scattering transform; diffeomorphism; Bäcklund transformation; hyperbolic manifold; compatibility conditions; generalized wave equation; generalized sine Gordon equation PDFBibTeX XMLCite \textit{M. J. Ablowitz} et al., Stud. Appl. Math. 74, 177--203 (1986; Zbl 0626.35082) Full Text: DOI References: [1] Gardner, Phys. Rev. Lett. 19: pp 1095– (1967) · doi:10.1103/PhysRevLett.19.1095 [2] Ablowitz, Solitons and the inverse scattering transform, SIAM Appl. Math. 4 (1981) · Zbl 0472.35002 [3] Ablowitz, Phys. Rev. Lett. 30: pp 1262– (1973) · doi:10.1103/PhysRevLett.30.1262 [4] Bäcklund, Concerning Surfaces with Constant Negative Curvature (1905) [5] Bianchi, L. Differenciale (1927) [6] Chern, Rocky Mountain Math. 10: pp 105– (1980) · Zbl 0407.53002 · doi:10.1216/RMJ-1980-10-1-105 [7] Tenenblat, Ann. of Math. 111 pp 477– (1980) · Zbl 0462.35079 · doi:10.2307/1971105 [8] Terng, Ann. of Math. 111: pp 491– (1980) · Zbl 0447.53001 · doi:10.2307/1971106 [9] Tenenblat, Bäcklund’s theorem for submanifolds of space forms, and generalized wave equations, Boletin da Sociedade Brasileira de Matematica 16 (1985) · Zbl 0618.53017 [10] Proceedings of the CIFMO School and Workshop held at Oaxtepec 189 (1982) 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.