zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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.

35Q58Other completely integrable PDE (MSC2000)
37J35Completely integrable systems, topological structure of phase space, integration methods
37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
76A15Liquid crystals (fluid mechanics)
Full Text: DOI
[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. · 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?. What is integrability?, 1 (1991) · 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. · Zbl 0808.76011
[6] Dorfman, I.: Dirac structures and integrability of nonlinear evolution equations. (1993) · 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). · Zbl 0856.35075
[8] Hunter, J. K.; Saxton, R. A.: Dynamics of director fields. SIAM J. Appl. math. 51, No. 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. · 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. · Zbl 0834.35085
[11] Olver, P.: Applications of Lie groups to differential equations. (1986) · Zbl 0588.22001
[12] P. Rosenau, Nonlinear dispersion and compact structures, Phys. Rev. Lett., submitted. · Zbl 0953.35501