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

*(English)*Zbl 0808.35124The following equation governing propagation of small-amplitude waves on a surface of a shallow layer of water is considered:
\[
U_ t- U_{xxt}+ kU_ x+ 3UU_ x= 2U_ x U_{xx}+ UU_{xxx}. \tag{1}
\]
Unlike the famous KdV equation, equation (1) does not assume that the wave length is much greater than the depth of water. The parameter \(k\) is related to a critical wave speed in the shallow water layer, while the variable \(U(x,t)\) represents the \(x\)-component of the velocity. Equation (1) is integrable, possessing a Lax pair and a bi-Hamiltonian structure. Recently, the particular case \(k=0\) was studied in detail. It was shown that in this case equation (1) gives rise to the so-called peakons (peaked solitons). In this work, an objective is to construct classes of exact solutions describing multipeakon states, as well as some other solution, e.g., solitons on a quasiperiodic background. The approach to the problem is based not on the inverse scattering technique, but rather on a technique of complex geometry. The integrable PDE (1) is reduced to integrable finite-dimensional Hamiltonian systems, and then special solutions to the PDE are expressed through solutions to the finite- dimensional systems. This approach allows to obtain, e.g., exact solutions describing a collision of two solitons. For this case, explicit formulas giving phase shifts produced by the collision are obtained (they prove to be essentially different from the well-known analogous formulas for the KdV solitons). Using a special limiting procedure, the so-called billiard solutions to the PDE (1) are obtained, too. These are exact solutions which remain finite everywhere along with the first derivative, but the derivative suffers discontinuities at certain points. It is stated that this is the first example of the billiard solutions which are obtained as weak solutions to an integrable nonlinear PDE.

Reviewer: B.A.Malomed (Ramat Aviv)

##### MSC:

35Q53 | KdV equations (Korteweg-de Vries equations) |

37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |

37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |

76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |

PDF
BibTeX
XML
Cite

\textit{M. S. Alber} et al., Lett. Math. Phys. 32, No. 2, 137--151 (1994; Zbl 0808.35124)

Full Text:
DOI

##### References:

[1] | Ablowitz, M. J. and Segur, H.,Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, 1981. · Zbl 0472.35002 |

[2] | 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). · Zbl 0413.35064 · doi:10.1112/jlms/s2-19.3.467 |

[3] | Alber, M. S., On integrable systems and semiclassical solutions of the stationary Schrödinger equations,Inverse Problems 5, 131-148 (1989). · Zbl 0691.35080 · doi:10.1088/0266-5611/5/2/003 |

[4] | 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). |

[5] | 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). · Zbl 0609.58011 · doi:10.1112/jlms/s2-36.1.176 |

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

[7] | Alber, M. S. and Marsden, J. E., On geometric phases for soliton equations,Comm. Math. Phys. 149, 217-240 (1992). · Zbl 0811.35115 · doi:10.1007/BF02097623 |

[8] | Alber, M. S. and Marsden, J. E.,Geometric Phases and Monodromy at Singularities, NATO Advanced Study Institute, Series C 1994, to appear. · Zbl 0852.35121 |

[9] | Alber, M. S. and Marsden, J. E., Resonant geometric phases for soliton equations,Fields Institute Commun. 1994, to appear. · Zbl 0830.58015 |

[10] | Camassa, R. and Holm, D. D., An integrable shallow water equation with peaked solitons,Phys. Rev. Lett,71, 1661-1664 (1993). · Zbl 0936.35153 · doi:10.1103/PhysRevLett.71.1661 |

[11] | Camassa, R., Holm, D. D., and Hyman, J. M., A new integrable shallow water equation,Adv. Appl. Mech. (1993), to appear. · Zbl 0808.76011 |

[12] | 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). · Zbl 0714.35074 · doi:10.1007/BF01234429 |

[13] | 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). · Zbl 1109.35374 · doi:10.1143/PTP.55.438 |

[14] | Ge, Z., Kruse, H. P., Marsden, J. E., and Scovel, C., Poisson brackets in the shallow water approximation, preprint (1993). · Zbl 0852.35113 |

[15] | 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). · Zbl 0351.76014 · doi:10.1017/S0022112076002425 |

[16] | Kruskal, M. D., Nonlinear wave equations, in J. Moser (ed),Dynamical Systems, Theory and Applications, Lecture Notes in Physics 38, Springer, New York, 1975. · Zbl 0322.35056 |

[17] | 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. · Zbl 0694.70008 |

[18] | McKean, H. P.,Integrable Systems and Algebraic Curves, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1979. · Zbl 0449.35080 |

[19] | McKean, H. P., Theta functions, solitons, and singular curves, in C. I. Byrnes (ed),PDE and Geometry, Proc. of Park City Conference, 1977. · Zbl 0411.58023 |

[20] | Morse, P. M. and Feshbach, H.,Methods of Theoretical Physics, McGraw-Hill, New York, 1953. · Zbl 0051.40603 |

[21] | Whitham, G. B.,Linear and Nonlinear Waves, Wiley, New York, 1974, p. 585. · Zbl 0373.76001 |

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

[23] | Weinstein, A., Connections of Berry and Hannay type for moving Lagrangian submanifolds,Adv. in Math. 82, 133-159 (1990). · Zbl 0713.58015 · doi:10.1016/0001-8708(90)90086-3 |

[24] | Wadati, M., Ichikawa, Y. H., and Shimizu, T., Cusp soliton of a new integrable nonlinear evolution equation,Prog. Theoret. Phys. 64, 1959-1967 (1980). · Zbl 1059.37506 · doi:10.1143/PTP.64.1959 |

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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.