##
**On a completely integrable nonlinear hyperbolic variational equation.**
*(English)*
Zbl 0900.35387

Summary: We show that the nonlinear partial differential equation, \((u_t+uu_x)_{xx}= 1/2(u_x^2)_x\), is a completely integrable, bi-variational, bi-Hamiltonian system. The corresponding equation for \(w= u_{xx}\) belongs to the Harry Dym hierarchy. This equation arises in two different physical contexts in two nonequivalent variational forms. It describes the propagation of weakly nonlinear orientation waves in a massive nematic liquid crystal director field, and it is the high-frequency limit of the Camassa-Holm equation, which is an integrable model equation for shallow water waves. Using the bi-Hamiltonian structure, we derive a recursion operator, a Lax pair, and an infinite family commuting Hamiltonian flows, together with the associated conservation laws. We also give the transformation to action-angle coordinates. Smooth solutions of the partial differential equation break down because their derivative blows up in finite time. Nevertheless, the Hamiltonian structure and complete integrability appear to remain valid globally in time, even after smooth solutions break down. We show this fact explicitly for finite dimensional invariant manifolds consisting of conservative piecewise linear solutions. We compute the bi-Hamiltonian structure on this invariant manifold, which is obtained by restricting the bi-Hamiltonian structure of the partial differential equation.

### MSC:

35Q58 | Other completely integrable PDE (MSC2000) |

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.) |

76A15 | Liquid crystals |

PDFBibTeX
XMLCite

\textit{J. K. Hunter} and \textit{Y. Zheng}, Physica D 79, No. 2--4, 361--386 (1994; Zbl 0900.35387)

Full Text:
DOI

### References:

[1] | M.S. Alber, R. Camassa, D.D. Holm and J.E. Marsden, The geometry of peaked solitons and billiard solutions of a class of integrable PDE’s, Lett. Math. Phys., to appear.; M.S. Alber, R. Camassa, D.D. Holm and J.E. Marsden, The geometry of peaked solitons and billiard solutions of a class of integrable PDE’s, Lett. Math. Phys., to appear. · Zbl 0808.35124 |

[2] | Calogero, F., A solvable nonlinear wave equation, Stud. Appl. Math., 70, 189 (1984) · Zbl 0551.35056 |

[3] | Calogero, F., Why are certain nonlinear PDE’s both widely applicable and integrable?, (Zakharov, V. E., What is Integrability? (1991), Springer: Springer Berlin), 1 · Zbl 0808.35001 |

[4] | Camassa, R.; Holm, D., An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71, 1661 (1993) · Zbl 0972.35521 |

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

[6] | Dorfman, I., (Dirac Structures and Integrability of Nonlinear Evolution Equations (1993), Wiley: Wiley Chichester) · Zbl 0717.58026 |

[7] | J.K. Hunter, Asymptotic equations for nonlinear hyperbolic waves, in: Surveys in Applied Mathematics, Vol. 2, J.B. Keller, D.W. McLaughlin and G. Papanicolaou, eds. (Plenum Press, New York).; J.K. Hunter, Asymptotic equations for nonlinear hyperbolic waves, in: Surveys in Applied Mathematics, Vol. 2, J.B. Keller, D.W. McLaughlin and G. Papanicolaou, eds. (Plenum Press, New York). · Zbl 0856.35075 |

[8] | Hunter, J. K.; Saxton, R. A., Dynamics of director fields, SIAM J. Appl. Math., 51, 6, 1498 (1991) · Zbl 0761.35063 |

[9] | J.K. Hunter and Y. Zheng, On a nonlinear hyperbolic variational equation: I. Global existence of weak solutions, Arch. Rat. Mech. Anal., to appear.; J.K. Hunter and Y. Zheng, On a nonlinear hyperbolic variational equation: I. Global existence of weak solutions, Arch. Rat. Mech. Anal., to appear. · Zbl 0834.35085 |

[10] | J.K. Hunter and Y. Zheng, On a nonlinear hyperbolic variational equation: II. Zero dispersion and dissipation limits, Arch. Rat. Mech. Anal., to appear.; J.K. Hunter and Y. Zheng, On a nonlinear hyperbolic variational equation: II. Zero dispersion and dissipation limits, Arch. Rat. Mech. Anal., to appear. · Zbl 0834.35085 |

[11] | Olver, P., (Applications of Lie Groups to Differential Equations (1986), Springer: Springer New York) |

[12] | P. Rosenau, Nonlinear dispersion and compact structures, Phys. Rev. Lett., submitted.; P. Rosenau, Nonlinear dispersion and compact structures, Phys. Rev. Lett., submitted. · Zbl 0953.35501 |

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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.