On complex-valued solutions to a 2D Eikonal equation. Part three: analysis of a Bäcklund transformation

References

• Kline , M . 1951 . An asymptotic solution of Maxwell's equation . Communications on Pure and Applied Mathematics , 4 : 225 – 262 .
   Web of Science ®Google Scholar
• Lewis , RM and Keller , JB . 1964 . “ Asymptotic methods for partial differential equations: the reduced wave equation and Maxwell's equation ” . New York : New York University, Courant Institute of Mathematical Sciences .
   Google Scholar
• Luneburg , RK . 1964 . Mathematical Theory of Optics , Berkeley, CA : University of California Press .
   Google Scholar
• Bouche , D and Molinet , F . 1994 . Méthodes Asymptotiques en Électromagnétisme , Berlin : Springer-Verlag .
   Google Scholar
• Choudhary , S and Felsen , LB . 1973 . Asymptotic theory for inhomogeneous waves . Institute of Electrical and Electronics Engineers. Transactions on Antennas and Propagation , 21 : 827 – 842 .
   Web of Science ®Google Scholar
• Choudhary , S and Felsen , LB . 1974 . Analysis of Gaussian beam propagation and diffraction by inhomogeneous wave tracking . Proceedings of IEEE , 62 : 1530 – 1541 .
   Web of Science ®Google Scholar
• Chapman , SJ , Lawry , JMH , Ockendon , JR and Tew , RH . 1999 . On the theory of complex rays . SIAM Review , 41 : 417 – 509 .
   Web of Science ®Google Scholar
• Einzinger , P and Felsen , LB . 1982 . Evanescent waves and complex rays . Institute of Electrical and Electronics Engineers. Transactions on Antennas and Propagation AP-30 , 4 : 594 – 605 .
   Google Scholar
• Einzinger , P and Raz , S . 1980 . On the asymptotic theory of inhomogeneous wave tracking . Radio Science , 15 : 763 – 771 .
   Web of Science ®Google Scholar
• Felsen , LB . 1976 . Evanescent waves . Journal of the Optical Society of America , 66 : 751 – 760 .
   Web of Science ®Google Scholar
• Felsen , LB . 1976 . Complex-source-point solutions of the field equations and their relation to the propagation and scattering of Gaussian beams . Symposia Mathematica , 18 : 39 – 56 .
   Google Scholar
• Heyman , E and Felsen , LB . 1983 . Evanescent waves and complex rays for modal propagation in curved open waveguides . SIAM Journal on Applied Mathematics. , 43 : 855 – 884 .
   Web of Science ®Google Scholar
• Kravtsov , YuA . 1964 . A modification of the geometrical optics method . Radiofizika , 7 : 664 – 673 .
   Google Scholar
• Kravtsov , YuA . 1964 . Asymptotic solutions of Maxwell's equations near a caustic . Radiofizika , 7 : 1049 – 1056 .
   Google Scholar
• Ludwig , D . 1966 . Uniform asymptotic expansions at a caustic . Communications on Pure and Applied Mathematics , 19 : 215 – 250 .
   Web of Science ®Google Scholar
• Ludwig , D . 1967 . Uniform asymptotic expansion of the field scattered by a convex object at high frequencies . Communications on Pure and Applied Mathematics , 20 : 103 – 138 .
   Web of Science ®Google Scholar
• Lewis , RM , Bleistein , N and Ludwig , D . 1967 . Uniform asymptotic theory of creeping waves . Communications on Pure and Applied Mathematics , 20 : 295 – 328 .
   Web of Science ®Google Scholar
• Anderson , RL and Ibragimov , NH . 1979 . “ Lie-Bäcklund Transformations in Applications ” . In Studies in Applied Mathematics , Philadelphia, PA : SIAM .
   Google Scholar
• Rogers , C and Shadwik , WF . 1982 . “ Mathematics in Science and Engineering ” . In Bäcklund Transformations and Their Applications , Vol. 161 , New York, London : Academic Press .
   Google Scholar
• Rogers , C , Schief , WK and Johnston , ME . 2000 . “ Bäcklund and his works: applications in soliton theory ” . In Geometric Approaches to Differential Equations , Edited by: Vassiliou , PJ and Lisle , IG . 16 – 55 . Princeton, NJ : Cambridge University Press . (Eds)
   Google Scholar
• Zwillinger , D . 1984 . Handbook of Differential Equations , Boston, MA : Academic Press .
   Google Scholar
• Rockafellar , RT . 1970 . Convex Analysis , Princeton, NJ : Princeton University Press .
   Google Scholar
• Aronsson , G . 1988 . On certain p-harmonic functions in the plane . Manuscripta Mathematica , 61 : 79 – 101 .
   Web of Science ®Google Scholar
• Aronsson , G . 1989 . Representation of a p-harmonic function near a critical point in the plane . Manuscripta Mathematica , 66 : 73 – 95 .
   Google Scholar
• Aronsson , G and Lindqvist , P . 1988 . On p-harmonic functions in the plane and their stream functions . Journal of Differential Equations , 74 : 157 – 178 .
   Web of Science ®Google Scholar
• Magnanini , R and Talenti , G . 1999 . On complex-valued solutions to a 2D Eikonal equation. Part One: qualitative properties . Contemporary Mathematics , 283 : 203 – 229 .
   Google Scholar
• Magnanini , R and Talenti , G . 2002 . “ Approaching a partial differential equation of mixed elliptic-hyperbolic type ” . In Ill-posed and Inverse Problems , Edited by: Romanov , VG . 263 – 276 . Zeist : VSP 2002 . et al. (Eds)
   Google Scholar
• Evans , LC . 1978 . A convergence theorem for solutions of nonlinear second order elliptic equations . Indiana University Mathematics Journal , 27 : 875 – 887 .
   Google Scholar
• Evans , LC . 1980 . On solving certain nonlinear partial differential equations by accretive operator methods . Israel Journal of Mathematics , 36 ( 3–4 ) : 225 – 247 .
   Google Scholar
• Crandall , MG , Evans , LC and Lions , PL . 1984 . Some properties of viscosity solutions of Hamilton–Jacobi equations . Transactions of the American Mathematical Society , 282 : 487 – 502 .
   Web of Science ®Google Scholar
• Crandall , MG and Lions , PL . 1983 . Viscosity solutions of Hamilton–Jacobi equations . Transactions of the American Mathematical Society , 277 : 1 – 42 .
   Web of Science ®Google Scholar
• Ziemer , WP . 1989 . Weakly Differentiable Functions , Berlin : Springer-Verlag .
   Google Scholar
• Crandall , MG , Ishii , H and Lions , PL . 1992 . User's guide to viscosity solutions of second order partial differential equations . Bulletin of American Mathematical Society , 27 : 1 – 67 .
   Web of Science ®Google Scholar
• Evans , LC . 1998 . Partial Differential Equations, Graduate Studies in Mathematics , Vol. 19 , Providence, RI : American Mathematical Society .
   Google Scholar
• Buseman , A . 1942 . Die achsensymmetrische kegelige überschallströmung . Luftfahrtforshung , 19 : 137 – 144 .
   Google Scholar
• Xu , ZF , Xu , ZY and Chen , GL . 1987 . Some notes on the Buseman equation . Advances in Mathematics (Chinese) (Beijing) , 16 : 81 – 86 . MR887969 (88e:35083)
   Google Scholar
• Magnanini , R and Talenti , G . 2002 . On complex-valued solutions to a 2D Eikonal equation. Part two: existence theorems . SIAM Journal on Mathematical Analysis , 34 : 805 – 835 .
   Google Scholar