Abstract
The real, nonsingular elliptic solutions of the Korteweg-de Vries equation are studied through the time dynamics of their poles in the complex plane. The dynamics of these poles is governed by a dynamical system with a constraint. This constraint is solvable for any finite number of poles located in the fundamental domain of the elliptic function, often in many different ways. Special consideration is given to those elliptic solutions that have a real nonsingular soliton limit.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ablowitz, M. J. and Segur, H.: Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, PA, 1981.
Airault, H., McKean, H. P., and Moser, J.: Rational and elliptic solutions of the Korteweg-de Vries equation and a related many-body problem, Comm. Pure Appl. Math. 30(1) (1977), 95–148.
Belokolos, E. D., Bobenko, A. I., Enol'skii, V. Z., Its, A. R., and Matveev, V. B.: Algebro-Geometric Approach to Nonlinear Integrable Problems, Springer Ser. Nonlinear Dynam., Springer-Verlag, Berlin, 1994.
Chudnovs'ki, D. V. and Chudnovs'ki, G. V.: Pole expansions of nonlinear partial differentialequations, Nuovo Cimento B (11) 40(2) (1977), 339–353.
Conway, J. B.: Functions of One Complex Variable, 2nd edn,Springer-Verlag, New York, 1978.
Dubrovin, B. A.: Theta functions and nonlinear equations, Russian Math. Surveys 36(2) (1981), 11–80.
Dubrovin, B. A. and Novikov, S. P.: Periodic and conditionally periodic analogs of the manysolitonsolutions of the Korteweg-de Vries equation, Soviet Phys. JETP 40 (1975), 1058–1063.
Ènols'kii, V. Z.: On solutions in elliptic functions of integrable nonlinear equations associatedwith two-zone Lamé potentials, Soviet Math. Dokl. 30 (1984), 394–397.
Gardner, C. S., Greene, J. M., Kruskal, M. D., and Miura, R. M.: Method for solving theKorteweg-de Vries equation, Phys. Rev. Lett. 19 (1967), 1095–1097.
Gesztesy, F. and Weikard, R.: On Picard potentials, Differential Integral Equations 8(6) (1995), 1453–1476.
Gesztesy, F. and Weikard, R.: Picard potentials and Hill's equation on a torus, Acta Math. 176(1) (1996), 73–107.
Gradshteyn, I. S. and Ryzhik, I. M.: Table of Integrals, Series, and Products, 5th edn, Translation edited and with a preface by Alan Jeffrey, Academic Press, Boston, MA, 1994.
Ince, E. L.: Further investigations into the periodic Lamé functions, Proc. Roy. Soc. Edinburgh 60 (1940), 83–99.
Its, A. R. and Matveev, V. B.: Schrödinger operators with the finite-band spectrum and the N soliton solutions of the Korteweg-de Vries equation, Theoret. and Math. Phys. 23(1) (1976),343–355.
Kruskal, M. D.: The Korteweg-de Vries equation and related evolution equations, Lectures in Appl. Math. 15, Amer. Math. Soc. Providence, 1974, pp. 61-83.
Lax, P. D.: Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math. 21 (1968), 467–490.
Thickstun, W. R.: A system of particles equivalent to solitons, J. Math. Anal. Appl. 55(2) (1976), 335–346.
Treibich, A. and Verdier, J.-L.: Revêtements tangentiels et sommes de 4 nombres triangulaires, C.R. Acad. Sci. Paris Sér. I Math. 311(1) (1990), 51–54.
Treibich, A. and Verdier, J.-L.: Solitons elliptiques, In: The Grothendieck Festschrift, Vol. III, With an appendix by J. Oesterlé, Birkhäuser, Boston, 1990, pp. 437–480.
Treibich, A. and Verdier, J.-L.: Revêtements exceptionnels et sommes de 4 nombres triangulaires, Duke Math. J. 68(2) (1992), 217–236.
Verdier, J.-L.: New elliptic solitons, In: Algebraic Analysis, Vol. II, Academic Press, Boston, MA, 1988, pp. 901–910.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Deconinck, B., Segur, H. Pole Dynamics for Elliptic Solutions of the Korteweg-deVries Equation. Mathematical Physics, Analysis and Geometry 3, 49–74 (2000). https://doi.org/10.1023/A:1009830803696
Issue Date:
DOI: https://doi.org/10.1023/A:1009830803696