Continuum limit of lattice approximation schemes

Journal of Mathematical Physics

Volumn 38, Issue 7, 1997, Pages 3700-3717

  Bender, Carl M ^a   Tovbis, Alexander ^b  

^a WASHINGTON UNIVERSITY   (United States)

^b UNIVERSITY OF CENTRAL FLORIDA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0031496832     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.532063     Document Type: Article

References (33)

1. C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978), Chap. 9.
2. C. M. Bender, F. Cooper, G. S. Guralnik, E. Mjolsness, H. Rose, and D. H. Sharp, Adv. App. Math. 1, 22 (1980).
3. C. M. Bender, Los Alamos Sci. 2, 76 (1981).
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6. C. M. Bender, F. Cooper, G. S. Guralnik. H. Moreno, R. Roskies, and D. H. Sharp, Phys. Rev. Lett. 45, 501 (1980); C. M. Bender, F. Cooper, G. S. Guralnik, R. Roskies, and D. H. Sharp, Phys. Rev. D 23, 2976 (1981), 23, 2999 (1981), and 24, 2683 (1981); C. M. Bender, F. Cooper, and A. Das, Phys. Rev. Lett. 50, 397 (1983); C. M. Bender, A. Das, H.-A. Lim, and L. M. Simmons, Jr., Phys. Lett. 134B, 225 (1984); C. M. Bender, A. Das, and H.-A. Lim, Pramana 23, 695 (1984); C. M. Bender, P. H. Burchard, A. Das, H.-A. Lim, and J. A. Shapiro, Phys. Rev. Lett. 54, 2481 (1985); C. M. Bender, A. Das, H.-A. Lim, and L. M. Simmons, Jr., Phys. Lett. 154B, 381 (1985); C. M. Bender, S. Boettcher, and L. Lipatov, Phys. Rev. Lett. 68, 3674 (1992) and Phys. Rev. D 46, 5557 (1992); C. M. Bender and S. Boettcher, Phys. Rev. D 48, 4919 (1993) and 51, 1875 (1994).
7. C. M. Bender, F. Cooper, G. S. Guralnik. H. Moreno, R. Roskies, and D. H. Sharp, Phys. Rev. Lett. 45, 501 (1980); C. M. Bender, F. Cooper, G. S. Guralnik, R. Roskies, and D. H. Sharp, Phys. Rev. D 23, 2976 (1981), 23, 2999 (1981), and 24, 2683 (1981); C. M. Bender, F. Cooper, and A. Das, Phys. Rev. Lett. 50, 397 (1983); C. M. Bender, A. Das, H.-A. Lim, and L. M. Simmons, Jr., Phys. Lett. 134B, 225 (1984); C. M. Bender, A. Das, and H.-A. Lim, Pramana 23, 695 (1984); C. M. Bender, P. H. Burchard, A. Das, H.-A. Lim, and J. A. Shapiro, Phys. Rev. Lett. 54, 2481 (1985); C. M. Bender, A. Das, H.-A. Lim, and L. M. Simmons, Jr., Phys. Lett. 154B, 381 (1985); C. M. Bender, S. Boettcher, and L. Lipatov, Phys. Rev. Lett. 68, 3674 (1992) and Phys. Rev. D 46, 5557 (1992); C. M. Bender and S. Boettcher, Phys. Rev. D 48, 4919 (1993) and 51, 1875 (1994).
8. C. M. Bender, F. Cooper, G. S. Guralnik. H. Moreno, R. Roskies, and D. H. Sharp, Phys. Rev. Lett. 45, 501 (1980); C. M. Bender, F. Cooper, G. S. Guralnik, R. Roskies, and D. H. Sharp, Phys. Rev. D 23, 2976 (1981), 23, 2999 (1981), and 24, 2683 (1981); C. M. Bender, F. Cooper, and A. Das, Phys. Rev. Lett. 50, 397 (1983); C. M. Bender, A. Das, H.-A. Lim, and L. M. Simmons, Jr., Phys. Lett. 134B, 225 (1984); C. M. Bender, A. Das, and H.-A. Lim, Pramana 23, 695 (1984); C. M. Bender, P. H. Burchard, A. Das, H.-A. Lim, and J. A. Shapiro, Phys. Rev. Lett. 54, 2481 (1985); C. M. Bender, A. Das, H.-A. Lim, and L. M. Simmons, Jr., Phys. Lett. 154B, 381 (1985); C. M. Bender, S. Boettcher, and L. Lipatov, Phys. Rev. Lett. 68, 3674 (1992) and Phys. Rev. D 46, 5557 (1992); C. M. Bender and S. Boettcher, Phys. Rev. D 48, 4919 (1993) and 51, 1875 (1994).
9. C. M. Bender, F. Cooper, G. S. Guralnik. H. Moreno, R. Roskies, and D. H. Sharp, Phys. Rev. Lett. 45, 501 (1980); C. M. Bender, F. Cooper, G. S. Guralnik, R. Roskies, and D. H. Sharp, Phys. Rev. D 23, 2976 (1981), 23, 2999 (1981), and 24, 2683 (1981); C. M. Bender, F. Cooper, and A. Das, Phys. Rev. Lett. 50, 397 (1983); C. M. Bender, A. Das, H.-A. Lim, and L. M. Simmons, Jr., Phys. Lett. 134B, 225 (1984); C. M. Bender, A. Das, and H.-A. Lim, Pramana 23, 695 (1984); C. M. Bender, P. H. Burchard, A. Das, H.-A. Lim, and J. A. Shapiro, Phys. Rev. Lett. 54, 2481 (1985); C. M. Bender, A. Das, H.-A. Lim, and L. M. Simmons, Jr., Phys. Lett. 154B, 381 (1985); C. M. Bender, S. Boettcher, and L. Lipatov, Phys. Rev. Lett. 68, 3674 (1992) and Phys. Rev. D 46, 5557 (1992); C. M. Bender and S. Boettcher, Phys. Rev. D 48, 4919 (1993) and 51, 1875 (1994).
10. C. M. Bender, F. Cooper, G. S. Guralnik. H. Moreno, R. Roskies, and D. H. Sharp, Phys. Rev. Lett. 45, 501 (1980); C. M. Bender, F. Cooper, G. S. Guralnik, R. Roskies, and D. H. Sharp, Phys. Rev. D 23, 2976 (1981), 23, 2999 (1981), and 24, 2683 (1981); C. M. Bender, F. Cooper, and A. Das, Phys. Rev. Lett. 50, 397 (1983); C. M. Bender, A. Das, H.-A. Lim, and L. M. Simmons, Jr., Phys. Lett. 134B, 225 (1984); C. M. Bender, A. Das, and H.-A. Lim, Pramana 23, 695 (1984); C. M. Bender, P. H. Burchard, A. Das, H.-A. Lim, and J. A. Shapiro, Phys. Rev. Lett. 54, 2481 (1985); C. M. Bender, A. Das, H.-A. Lim, and L. M. Simmons, Jr., Phys. Lett. 154B, 381 (1985); C. M. Bender, S. Boettcher, and L. Lipatov, Phys. Rev. Lett. 68, 3674 (1992) and Phys. Rev. D 46, 5557 (1992); C. M. Bender and S. Boettcher, Phys. Rev. D 48, 4919 (1993) and 51, 1875 (1994).
11. C. M. Bender, F. Cooper, G. S. Guralnik. H. Moreno, R. Roskies, and D. H. Sharp, Phys. Rev. Lett. 45, 501 (1980); C. M. Bender, F. Cooper, G. S. Guralnik, R. Roskies, and D. H. Sharp, Phys. Rev. D 23, 2976 (1981), 23, 2999 (1981), and 24, 2683 (1981); C. M. Bender, F. Cooper, and A. Das, Phys. Rev. Lett. 50, 397 (1983); C. M. Bender, A. Das, H.-A. Lim, and L. M. Simmons, Jr., Phys. Lett. 134B, 225 (1984); C. M. Bender, A. Das, and H.-A. Lim, Pramana 23, 695 (1984); C. M. Bender, P. H. Burchard, A. Das, H.-A. Lim, and J. A. Shapiro, Phys. Rev. Lett. 54, 2481 (1985); C. M. Bender, A. Das, H.-A. Lim, and L. M. Simmons, Jr., Phys. Lett. 154B, 381 (1985); C. M. Bender, S. Boettcher, and L. Lipatov, Phys. Rev. Lett. 68, 3674 (1992) and Phys. Rev. D 46, 5557 (1992); C. M. Bender and S. Boettcher, Phys. Rev. D 48, 4919 (1993) and 51, 1875 (1994).
12. C. M. Bender, F. Cooper, G. S. Guralnik. H. Moreno, R. Roskies, and D. H. Sharp, Phys. Rev. Lett. 45, 501 (1980); C. M. Bender, F. Cooper, G. S. Guralnik, R. Roskies, and D. H. Sharp, Phys. Rev. D 23, 2976 (1981), 23, 2999 (1981), and 24, 2683 (1981); C. M. Bender, F. Cooper, and A. Das, Phys. Rev. Lett. 50, 397 (1983); C. M. Bender, A. Das, H.-A. Lim, and L. M. Simmons, Jr., Phys. Lett. 134B, 225 (1984); C. M. Bender, A. Das, and H.-A. Lim, Pramana 23, 695 (1984); C. M. Bender, P. H. Burchard, A. Das, H.-A. Lim, and J. A. Shapiro, Phys. Rev. Lett. 54, 2481 (1985); C. M. Bender, A. Das, H.-A. Lim, and L. M. Simmons, Jr., Phys. Lett. 154B, 381 (1985); C. M. Bender, S. Boettcher, and L. Lipatov, Phys. Rev. Lett. 68, 3674 (1992) and Phys. Rev. D 46, 5557 (1992); C. M. Bender and S. Boettcher, Phys. Rev. D 48, 4919 (1993) and 51, 1875 (1994).
13. C. M. Bender, F. Cooper, G. S. Guralnik. H. Moreno, R. Roskies, and D. H. Sharp, Phys. Rev. Lett. 45, 501 (1980); C. M. Bender, F. Cooper, G. S. Guralnik, R. Roskies, and D. H. Sharp, Phys. Rev. D 23, 2976 (1981), 23, 2999 (1981), and 24, 2683 (1981); C. M. Bender, F. Cooper, and A. Das, Phys. Rev. Lett. 50, 397 (1983); C. M. Bender, A. Das, H.-A. Lim, and L. M. Simmons, Jr., Phys. Lett. 134B, 225 (1984); C. M. Bender, A. Das, and H.-A. Lim, Pramana 23, 695 (1984); C. M. Bender, P. H. Burchard, A. Das, H.-A. Lim, and J. A. Shapiro, Phys. Rev. Lett. 54, 2481 (1985); C. M. Bender, A. Das, H.-A. Lim, and L. M. Simmons, Jr., Phys. Lett. 154B, 381 (1985); C. M. Bender, S. Boettcher, and L. Lipatov, Phys. Rev. Lett. 68, 3674 (1992) and Phys. Rev. D 46, 5557 (1992); C. M. Bender and S. Boettcher, Phys. Rev. D 48, 4919 (1993) and 51, 1875 (1994).
14. C. M. Bender, F. Cooper, G. S. Guralnik. H. Moreno, R. Roskies, and D. H. Sharp, Phys. Rev. Lett. 45, 501 (1980); C. M. Bender, F. Cooper, G. S. Guralnik, R. Roskies, and D. H. Sharp, Phys. Rev. D 23, 2976 (1981), 23, 2999 (1981), and 24, 2683 (1981); C. M. Bender, F. Cooper, and A. Das, Phys. Rev. Lett. 50, 397 (1983); C. M. Bender, A. Das, H.-A. Lim, and L. M. Simmons, Jr., Phys. Lett. 134B, 225 (1984); C. M. Bender, A. Das, and H.-A. Lim, Pramana 23, 695 (1984); C. M. Bender, P. H. Burchard, A. Das, H.-A. Lim, and J. A. Shapiro, Phys. Rev. Lett. 54, 2481 (1985); C. M. Bender, A. Das, H.-A. Lim, and L. M. Simmons, Jr., Phys. Lett. 154B, 381 (1985); C. M. Bender, S. Boettcher, and L. Lipatov, Phys. Rev. Lett. 68, 3674 (1992) and Phys. Rev. D 46, 5557 (1992); C. M. Bender and S. Boettcher, Phys. Rev. D 48, 4919 (1993) and 51, 1875 (1994).
15. C. M. Bender, F. Cooper, G. S. Guralnik. H. Moreno, R. Roskies, and D. H. Sharp, Phys. Rev. Lett. 45, 501 (1980); C. M. Bender, F. Cooper, G. S. Guralnik, R. Roskies, and D. H. Sharp, Phys. Rev. D 23, 2976 (1981), 23, 2999 (1981), and 24, 2683 (1981); C. M. Bender, F. Cooper, and A. Das, Phys. Rev. Lett. 50, 397 (1983); C. M. Bender, A. Das, H.-A. Lim, and L. M. Simmons, Jr., Phys. Lett. 134B, 225 (1984); C. M. Bender, A. Das, and H.-A. Lim, Pramana 23, 695 (1984); C. M. Bender, P. H. Burchard, A. Das, H.-A. Lim, and J. A. Shapiro, Phys. Rev. Lett. 54, 2481 (1985); C. M. Bender, A. Das, H.-A. Lim, and L. M. Simmons, Jr., Phys. Lett. 154B, 381 (1985); C. M. Bender, S. Boettcher, and L. Lipatov, Phys. Rev. Lett. 68, 3674 (1992) and Phys. Rev. D 46, 5557 (1992); C. M. Bender and S. Boettcher, Phys. Rev. D 48, 4919 (1993) and 51, 1875 (1994).
16. C. M. Bender, F. Cooper, G. S. Guralnik. H. Moreno, R. Roskies, and D. H. Sharp, Phys. Rev. Lett. 45, 501 (1980); C. M. Bender, F. Cooper, G. S. Guralnik, R. Roskies, and D. H. Sharp, Phys. Rev. D 23, 2976 (1981), 23, 2999 (1981), and 24, 2683 (1981); C. M. Bender, F. Cooper, and A. Das, Phys. Rev. Lett. 50, 397 (1983); C. M. Bender, A. Das, H.-A. Lim, and L. M. Simmons, Jr., Phys. Lett. 134B, 225 (1984); C. M. Bender, A. Das, and H.-A. Lim, Pramana 23, 695 (1984); C. M. Bender, P. H. Burchard, A. Das, H.-A. Lim, and J. A. Shapiro, Phys. Rev. Lett. 54, 2481 (1985); C. M. Bender, A. Das, H.-A. Lim, and L. M. Simmons, Jr., Phys. Lett. 154B, 381 (1985); C. M. Bender, S. Boettcher, and L. Lipatov, Phys. Rev. Lett. 68, 3674 (1992) and Phys. Rev. D 46, 5557 (1992); C. M. Bender and S. Boettcher, Phys. Rev. D 48, 4919 (1993) and 51, 1875 (1994).
17. C. M. Bender, F. Cooper, G. S. Guralnik. H. Moreno, R. Roskies, and D. H. Sharp, Phys. Rev. Lett. 45, 501 (1980); C. M. Bender, F. Cooper, G. S. Guralnik, R. Roskies, and D. H. Sharp, Phys. Rev. D 23, 2976 (1981), 23, 2999 (1981), and 24, 2683 (1981); C. M. Bender, F. Cooper, and A. Das, Phys. Rev. Lett. 50, 397 (1983); C. M. Bender, A. Das, H.-A. Lim, and L. M. Simmons, Jr., Phys. Lett. 134B, 225 (1984); C. M. Bender, A. Das, and H.-A. Lim, Pramana 23, 695 (1984); C. M. Bender, P. H. Burchard, A. Das, H.-A. Lim, and J. A. Shapiro, Phys. Rev. Lett. 54, 2481 (1985); C. M. Bender, A. Das, H.-A. Lim, and L. M. Simmons, Jr., Phys. Lett. 154B, 381 (1985); C. M. Bender, S. Boettcher, and L. Lipatov, Phys. Rev. Lett. 68, 3674 (1992) and Phys. Rev. D 46, 5557 (1992); C. M. Bender and S. Boettcher, Phys. Rev. D 48, 4919 (1993) and 51, 1875 (1994).
18. E. R. Caianello, Nuovo Cimento 10, 1634 (1953) and 11, 492 (1954); E. R. Caianello and G. Scarpetta, Nuovo Cimento A 22, 448 (1974), and Lett. Nuovo Cimento 12, 283 (1974).
19. E. R. Caianello, Nuovo Cimento 10, 1634 (1953) and 11, 492 (1954); E. R. Caianello and G. Scarpetta, Nuovo Cimento A 22, 448 (1974), and Lett. Nuovo Cimento 12, 283 (1974).
20. E. R. Caianello, Nuovo Cimento 10, 1634 (1953) and 11, 492 (1954); E. R. Caianello and G. Scarpetta, Nuovo Cimento A 22, 448 (1974), and Lett. Nuovo Cimento 12, 283 (1974).
21. E. R. Caianello, Nuovo Cimento 10, 1634 (1953) and 11, 492 (1954); E. R. Caianello and G. Scarpetta, Nuovo Cimento A 22, 448 (1974), and Lett. Nuovo Cimento 12, 283 (1974).
22. C M. Bender, F. Cooper, G. S. Guralnik, and D. H. Sharp, Phys. Rev. D 19, 1865 (1979).
23. C. M. Bender, F. Cooper, G. S. Guralnik, R. Roskies, and D. H. Sharp, Phys. Rev. Lett. 43, 537 (1979).
24. R. J. Rivers, Phys. Rev. D 20, 3425 (1979).
25. A rigorous proof of splitting separatrices in general can be found in A. Tovbis, Ann. Num. Math, (in press); for two-dimensional systems, see, V. F. Lazutkin, Funct. Anal. Ego Pril. 22, 83 (1988) (in Russian). Chaotic behavior of systems with a quadratic nonlinearity is demonstrated in A. Tovbis, Commun. Math. Phys. 163, 245 (1994); A. Tovbis, M. Tsuchiya, and C. Jaffe, Proceedings of the International Conference on Differential Equations and Applications to Biology and Industry (International, Singapore, 1995) A. Tovbis, M. Tsuchiya, and C. Jaffe, Advanced Studies Institute Series (Kluwer Academic, Dordrecht, in press); C. J. Amick and J. B. McLeod, SAACM, 1 127 (1991). The cubic case is considered in K. Nakamura and M. Hamada, submitted to J. Phys. A, and B. M. Herbst and M. J. Ablowitz (see Ref. 11).
26. A rigorous proof of splitting separatrices in general can be found in A. Tovbis, Ann. Num. Math, (in press); for two-dimensional systems, see, V. F. Lazutkin, Funct. Anal. Ego Pril. 22, 83 (1988) (in Russian). Chaotic behavior of systems with a quadratic nonlinearity is demonstrated in A. Tovbis, Commun. Math. Phys. 163, 245 (1994); A. Tovbis, M. Tsuchiya, and C. Jaffe, Proceedings of the International Conference on Differential Equations and Applications to Biology and Industry (International, Singapore, 1995) A. Tovbis, M. Tsuchiya, and C. Jaffe, Advanced Studies Institute Series (Kluwer Academic, Dordrecht, in press); C. J. Amick and J. B. McLeod, SAACM, 1 127 (1991). The cubic case is considered in K. Nakamura and M. Hamada, submitted to J. Phys. A, and B. M. Herbst and M. J. Ablowitz (see Ref. 11).
27. A rigorous proof of splitting separatrices in general can be found in A. Tovbis, Ann. Num. Math, (in press); for two-dimensional systems, see, V. F. Lazutkin, Funct. Anal. Ego Pril. 22, 83 (1988) (in Russian). Chaotic behavior of systems with a quadratic nonlinearity is demonstrated in A. Tovbis, Commun. Math. Phys. 163, 245 (1994); A. Tovbis, M. Tsuchiya, and C. Jaffe, Proceedings of the International Conference on Differential Equations and Applications to Biology and Industry (International, Singapore, 1995) A. Tovbis, M. Tsuchiya, and C. Jaffe, Advanced Studies Institute Series (Kluwer Academic, Dordrecht, in press); C. J. Amick and J. B. McLeod, SAACM, 1 127 (1991). The cubic case is considered in K. Nakamura and M. Hamada, submitted to J. Phys. A, and B. M. Herbst and M. J. Ablowitz (see Ref. 11).
28. A rigorous proof of splitting separatrices in general can be found in A. Tovbis, Ann. Num. Math, (in press); for two-dimensional systems, see, V. F. Lazutkin, Funct. Anal. Ego Pril. 22, 83 (1988) (in Russian). Chaotic behavior of systems with a quadratic nonlinearity is demonstrated in A. Tovbis, Commun. Math. Phys. 163, 245 (1994); A. Tovbis, M. Tsuchiya, and C. Jaffe, Proceedings of the International Conference on Differential Equations and Applications to Biology and Industry (International, Singapore, 1995) A. Tovbis, M. Tsuchiya, and C. Jaffe, Advanced Studies Institute Series (Kluwer Academic, Dordrecht, in press); C. J. Amick and J. B. McLeod, SAACM, 1 127 (1991). The cubic case is considered in K. Nakamura and M. Hamada, submitted to J. Phys. A, and B. M. Herbst and M. J. Ablowitz (see Ref. 11).
29. A rigorous proof of splitting separatrices in general can be found in A. Tovbis, Ann. Num. Math, (in press); for two-dimensional systems, see, V. F. Lazutkin, Funct. Anal. Ego Pril. 22, 83 (1988) (in Russian). Chaotic behavior of systems with a quadratic nonlinearity is demonstrated in A. Tovbis, Commun. Math. Phys. 163, 245 (1994); A. Tovbis, M. Tsuchiya, and C. Jaffe, Proceedings of the International Conference on Differential Equations and Applications to Biology and Industry (International, Singapore, 1995) A. Tovbis, M. Tsuchiya, and C. Jaffe, Advanced Studies Institute Series (Kluwer Academic, Dordrecht, in press); C. J. Amick and J. B. McLeod, SAACM, 1 127 (1991). The cubic case is considered in K. Nakamura and M. Hamada, submitted to J. Phys. A, and B. M. Herbst and M. J. Ablowitz (see Ref. 11).
30. A rigorous proof of splitting separatrices in general can be found in A. Tovbis, Ann. Num. Math, (in press); for two-dimensional systems, see, V. F. Lazutkin, Funct. Anal. Ego Pril. 22, 83 (1988) (in Russian). Chaotic behavior of systems with a quadratic nonlinearity is demonstrated in A. Tovbis, Commun. Math. Phys. 163, 245 (1994); A. Tovbis, M. Tsuchiya, and C. Jaffe, Proceedings of the International Conference on Differential Equations and Applications to Biology and Industry (International, Singapore, 1995) A. Tovbis, M. Tsuchiya, and C. Jaffe, Advanced Studies Institute Series (Kluwer Academic, Dordrecht, in press); C. J. Amick and J. B. McLeod, SAACM, 1 127 (1991). The cubic case is considered in K. Nakamura and M. Hamada, submitted to J. Phys. A, and B. M. Herbst and M. J. Ablowitz (see Ref. 11).
31. A rigorous proof of splitting separatrices in general can be found in A. Tovbis, Ann. Num. Math, (in press); for two-dimensional systems, see, V. F. Lazutkin, Funct. Anal. Ego Pril. 22, 83 (1988) (in Russian). Chaotic behavior of systems with a quadratic nonlinearity is demonstrated in A. Tovbis, Commun. Math. Phys. 163, 245 (1994); A. Tovbis, M. Tsuchiya, and C. Jaffe, Proceedings of the International Conference on Differential Equations and Applications to Biology and Industry (International, Singapore, 1995) A. Tovbis, M. Tsuchiya, and C. Jaffe, Advanced Studies Institute Series (Kluwer Academic, Dordrecht, in press); C. J. Amick and J. B. McLeod, SAACM, 1 127 (1991). The cubic case is considered in K. Nakamura and M. Hamada, submitted to J. Phys. A, and B. M. Herbst and M. J. Ablowitz (see Ref. 11).
32. Similar kinds of exact solutions to difference equations having cubic nonlinear terms have been found by B. M. Herbst and M. J. Ablowitz, J. Comput. Phys. 105, 122 (1993).
33. N for the continuum value of y(0) converge, but they do so in an interesting oscillatory fashion. We examine this convergence behavior in a separate paper.