The geometry of peaked solitons and billiard solutions of a class of integrable PDE's

Mark Alber1, Roberto Camassa2, Darryl D. Holm2, Jerrold E. Marsden3
1School of Mathematics, Institute for Advanced Study, Princeton and Department of Mathematics, University of Notre Dame, Notre Dame, USA
2Center for Nonlinear Studies and Theoretical Division, Los Alamos National Laboratory, Los Alamos, USA
3Department of Mathematics, University of California, Berkeley, USA

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Ablowitz, M. J. and Segur, H.,Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, 1981.

Alber, S. J., Investigation of equations of Koteweg-de Vries type by the method of recurrence relations,J. London Math. Soc. 19, 467?480 (1979).

Alber, M. S., On integrable systems and semiclassical solutions of the stationary Schrödinger equations,Inverse Problems 5, 131?148 (1989).

Alber, M. S. and Alber, S. J., Hamiltonian formalism for finite-zone solutions of integrable equations,C.R. Acad. Sci. Paris 301, 777?781 (1985).

Alber, M. S. and Alber, S. J., Hamiltonian formalism for nonlinear Schrödinger equations and sine-Gordon equations,J. London Math. Soc. 36, 176?192 (1987).

Alber, M. S., Camassa, R., Holm, D. D., and Marsden, J. E., in preparation (1994).

Alber, M. S. and Marsden, J. E., On geometric phases for soliton equations,Comm. Math. Phys. 149, 217?240 (1992).

Alber, M. S. and Marsden, J. E.,Geometric Phases and Monodromy at Singularities, NATO Advanced Study Institute, Series C 1994, to appear.

Alber, M. S. and Marsden, J. E., Resonant geometric phases for soliton equations,Fields Institute Commun. 1994, to appear.

Camassa, R. and Holm, D. D., An integrable shallow water equation with peaked solitons,Phys. Rev. Lett,71, 1661?1664 (1993).

Camassa, R., Holm, D. D., and Hyman, J. M., A new integrable shallow water equation,Adv. Appl. Mech. (1993), to appear.

Ercolani, N. and McKean, H. P., Geometry of KdV(4): Abel sums, Jacobi variety, and theta function in the scattering case,Invent. Math. 99, 483 (1990).

Flaschka, H. and McLaughlin, D. W., Canonically conjugate variables for the Korteweg-de Vries equation and the Toda lattice with periodic boundary conditions,Prog. Theoret. Phys. 55, 438?456 (1976).

Ge, Z., Kruse, H. P., Marsden, J. E., and Scovel, C., Poisson brackets in the shallow water approximation, preprint (1993).

Green, A. E. and Naghdi, P. M., A derivation of equations for wave propagation in water of variable depth,J. Fluid Mech. 78, 237?246 (1976).

Kruskal, M. D., Nonlinear wave equations, in J. Moser (ed),Dynamical Systems, Theory and Applications, Lecture Notes in Physics 38, Springer, New York, 1975.

Marsden, J. E., Montgomery, R., and Ratiu, T., Cartan-Hannay-Berry phases and symmetry,Contemp. Math. 97, 279 (1989); see alsoMem. Amer. Math. Soc. 436.

McKean, H. P.,Integrable Systems and Algebraic Curves, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1979.

McKean, H. P., Theta functions, solitons, and singular curves, in C. I. Byrnes (ed),PDE and Geometry, Proc. of Park City Conference, 1977.

Morse, P. M. and Feshbach, H.,Methods of Theoretical Physics, McGraw-Hill, New York, 1953.

Whitham, G. B.,Linear and Nonlinear Waves, Wiley, New York, 1974, p. 585.

Whitham, G. B., Notes from the course ?Special Topics in Nonlinear Wave Propagation?, California Institute of Technology, Pasadena CA, 1988.

Weinstein, A., Connections of Berry and Hannay type for moving Lagrangian submanifolds,Adv. in Math. 82, 133?159 (1990).

Wadati, M., Ichikawa, Y. H., and Shimizu, T., Cusp soliton of a new integrable nonlinear evolution equation,Prog. Theoret. Phys. 64, 1959?1967 (1980).