Revista Matemática Iberoamericana

Algebro-Geometric Solutions of the Camassa-Holm hierarchy

Fritz Gesztesy and Helge Holden

Full-text: Open access

Abstract

We provide a detailed treatment of the Camassa-Holm (CH) hierarchy with special emphasis on its algebro-geometric solutions. In analogy to other completely integrable hierarchies of soliton equations such as the KdV or AKNS hierarchies, the CH hierarchy is recursively constructed by means of a basic polynomial formalism invoking a spectral parameter. Moreover, we study Dubrovin-type equations for auxiliary divisors and associated trace formulas, consider the corresponding algebro-geometric initial value problem, and derive the theta function representations of algebro-geometric solutions of the CH hierarchy.

Article information

Source
Rev. Mat. Iberoamericana Volume 19, Number 1 (2003), 73-142.

Dates
First available in Project Euclid: 31 March 2003

Permanent link to this document
http://projecteuclid.org/euclid.rmi/1049123081

Mathematical Reviews number (MathSciNet)
MR1993416

Zentralblatt MATH identifier
1029.37049

Subjects
Primary: 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10] 58F07
Secondary: 35Q51: Soliton-like equations [See also 37K40]

Keywords
Camassa-Holm hierarchy algebro-geometric solutions Dubrovin equations trace formulas

Citation

Gesztesy, Fritz; Holden, Helge. Algebro-Geometric Solutions of the Camassa-Holm hierarchy. Rev. Mat. Iberoamericana 19 (2003), no. 1, 73--142. http://projecteuclid.org/euclid.rmi/1049123081.


Export citation

References

  • Alber, M.,S.: $N$-component integrable systems and geometric asymptotics. In Integrability: The Seiberg-Witten and Whitham equations (H.,W. Braden and I.,M. Krichever, editors). Gordon and Breach Science Publishers, Singapore, 2000, 213--228.
  • Alber, M.,S., Camassa, R., Fedorov, Yu.,N., Holm, D.,D. and Marsden, J.,E.: On billiard solutions of nonlinear PDE's. Phys. Lett. A 264 (1999), 171--178.
  • Alber, M.,S., Camassa, R., Fedorov, Yu.,N., Holm, D.,D. and Marsden, J.,E.: The complex geometry of weak piecewise smooth solutions of integrable nonlinear PDE's of shallow water and Dym type. Comm. Math. Phys. 221 (2001), 197--227.
  • Alber, M.,S., Camassa, R. and Gekhtman, M.: Billiard weak solutions of nonlinear PDE's and Toda flows. In: SIDE III--Symmetries and Integrability of Difference Equations (D. Levi and O. Ragnisco, editors). CRM Proceedings and Lecture Notes 25, 1--10. Amer. Math. Soc., Providence, RI, 2000.
  • Alber, M.,S., Camassa, R., Holm, D.,D. and Marsden, J.,E.: The geometry of peaked solitons and billiard solutions of a class of integrable PDE's. Lett. Math. Phys. 32 (1994), 137--151.
  • Alber, M.,S., Camassa, R., Holm, D.,D. and Marsden, J.,E.: On the link between umbilic geodesics and soliton solutions of nonlinear PDE's. Proc. Roy. Soc. London Ser. A 450 (1995), 677--692.
  • Alber, M.,S. and Fedorov, Yu.,N.: Wave solutions of evolution equations and Hamiltonian flows on nonlinear subvarieties of generalized Jacobians. J. Phys. A 33 (2000), 8409--8425.
  • Alber, M.,S. and Fedorov, Yu.,N.: Algebraic geometrical solutions for certain evolution equations and Hamiltonian flows on nonlinear subvarieties of generalized Jacobians. Inverse Problems 17 (2001), 1017--1042.
  • Alber, M.,S., Luther, G.,G. and Miller, C.,A.: On soliton-type solutions of equations associated with $N$-component systems. J. Math. Phys. 41 (2000), 284--316.
  • Alber, M.,S. and Miller, C.: Peakon solitons of the shallow water equation. Appl. Math. Lett 14 (2001), 93--98.
  • Beals, R., Sattinger, D.,H. and Szmigielski, J.: Acoustic scattering and the extended Korteweg--de Vries hierarchy. Adv. Math. 140 (1998), 190--206.
  • Beals, R., Sattinger, D.,H. and Szmigielski, J.: Multi-peakons and a theorem of Stieltjes. Inverse Problems 15 (1999), no. 1, L1--L4.
  • Beals, R., Sattinger, D.,H. and Szmigielski, J.: Multipeakons and the classical moment problem. Adv. in Math. 154 (2000), 229--257.
  • Beals, R., Sattinger, D.,H. and Szmigielski, J.: Peakons, strings, and the finite Toda lattice. Comm. Pure Appl. Math. 54 (2001), 91--106.
  • Belokolos, E.,D., Bobenko, A.,I., Enol'skii, V.,Z., Its, A.,R. and Matveev, V.,B.: Algebro-Geometric Approach to Nonlinear Integrable Equations. Springer, Berlin, 1994.
  • Bulla, W., Gesztesy, F., Holden, H. and Teschl, G.: Algebro-geometric quasi-periodic finite-gap solutions of the Toda and Kac-van Moerbeke hierarchy. Mem. Amer. Math. Soc. 135 (1998), no. 641, 1--79.
  • Camassa, R. and Holm, D.,D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71 (1993), 1661--1664.
  • Camassa, R., Holm, D.,D. and Hyman, J.,M.: A new integrable shallow water equation. Adv. Appl. Mech. 31 (1994), 1--33.
  • Clebsch, A. and Gordan, P.: Theorie der Abelschen Funktionen, Teubner, Leipzig, 1866.
  • Constantin, A.: On the Cauchy problem for the periodic Camassa--Holm equation. J. Differential Equations 141 (1997), 218--235.
  • Constantin, A.: On the inverse spectral problem for the Camassa--Holm equation. J. Funct. Anal. 155 (1998), 352--363.
  • Constantin, A.: Quasi-periodicity with respect to time of spatially periodic finite-gap solutions of the Camassa--Holm equation. Bull. Sci. Math.,122 (1998), 487--494.
  • Constantin, A.: Existence of permanent and breaking waves for a shallow water equation: a geometric approach. Ann. Inst. Fourier (Grenoble) 50 (2000), 321--362.
  • Constantin, A. and Escher, J.: Global existence and blow-up for a shallow water equation. Ann. Scuola Norm. Sup. Pisa Cl. Sci.(4) 26 (1998), 303--328.
  • Constantin, A. and Escher, J.: Global weak solutions for a shallow water equation. Indiana Univ. Math. J. 47 (1998), 1527--1545.
  • Constantin, A. and Escher, J.: Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. 181 (1998), 229--243.
  • Constantin, A. and Escher, J.: Well-posedness, global existence, and blow-up phenomena for a periodic quasi-linear hyperbolic equation. Comm. Pure Appl. Math. 51 (1998), 475--504.
  • Constantin, A. and Escher, J.: On the blow-up rate and the blow-up set of breaking waves for a shallow water equation. Math. Z. 233 (2000), 75--91.
  • Constantin, A. and McKean, H.,P.: A shallow water equation on the circle. Comm. Pure Appl. Math. 52 (1999), 949--982.
  • Constantin, A. and Molinet, L.: Global weak solutions for a shallow water equation. Comm. Math. Phys. 211 (2000), 45--61.
  • Dickson, R., Gesztesy, F. and Unterkofler, K.: Algebro-geometric solutions of the Boussinesq hierarchy. Rev. Math. Phys. 11 (1999), 823--879.
  • Dickson, R., Gesztesy, F. and Unterkofler, K.: A new approach to the Boussinesq hierarchy. Math. Nachr. 198 (1999), 51--108.
  • Dmitrieva, L.,A.: Finite-gap solutions of the Harry Dym equation. Phys. Lett. A 182 (1993), 65--70.
  • Dullin, H.,R., Gottwald, G. and Holm, D.,D.: An integrable shallow water equation with linear and nonlinear dispersion. Phys. Rev. Lett. 87 (2001), 4501--4504.
  • Enolskii, V.,Z., Gesztesy, F. and Holden, H.: The classical massive Thirring system revisited. In: Stochastic Processes, Physics and Geometry: New Interplays. I: A Volume in Honor of Sergio Albeverio (F. Gesztesy, H. Holden, J. Jost, S. Paycha, M.Röckner, and S. Scarlatti, editors). CMS Conference Proceedings 28, Amer. Math. Soc., Providence, RI, 2000, 163--200.
  • Farkas, H.,M. and Kra, I.: Riemann Surfaces, second edition, Springer, New York, 1992.
  • Fedorov, Yu.: Classical integrable systems and billiards related to generalized Jacobians. Acta Appl. Math. 55 (1999), 251--301.
  • Fisher, M. and Schiff, J.: The Camassa Holm equation: conserved quantities and the initial value problem. Phys.,Lett.,A 259 (1999), 371--376.
  • Foias, C., Holm, D.,D. and Titi, E.,S.: The three dimensional viscous Camassa--Holm equations, and their relation to the Navier--Stokes equations and turbulence theory. J. Dynam. Diff. Eq. 14 (2002), 1--35.
  • Fornberg, B. and Witham, G.,B.: A numerical and theoretical study of certain nonlinear wave phenomena. Philos. Trans. Roy. Soc. London Ser. A 289 (1978), 373--404.
  • Fuchssteiner, B.: Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa--Holm equation. Phys. D 95 (1996), 229--243.
  • Fuchssteiner, B. and Fokas, A.,S.: Symplectic structures, their Bäcklund transformations and hereditary symmetries. Phys. D 4 (1981), 47--66.
  • Gagnon, L., Harnad, J., Winternitz, P. and Hurtubise, J.: Abelian integrals and the reduction method for an integrable Hamiltonian system. J. Math. Phys. 26 (1985), 1605--1612.
  • Gavrilov, L.: Generalized Jacobians of spectral curves and completely integrable systems. Math. Z. 230 (1999), 487--508.
  • Gesztesy, F. and Holden, H.: Dubrovin equations and integrable systems on hyperelliptic curves. Math. Scand. 91 (2002), 91--126.
  • Gesztesy, F. and Holden, H.: Soliton Equations and Their Algebro--Geometric Solutions. Vol. I: $(1+1)$-Dimensional Continuous Models. Cambridge Studies in Advanced Mathematics, Cambridge Univ. Press, Cambridge, 2003.
  • Gesztesy, F. and Holden, H.: A combined sine-Gordon and modified Korteweg--de Vries hierarchy and its algebro-geometric solutions. In: Differential Equations and Mathematical Physics (R. Weikard and G. Weinstein, editors), 133--173. Studies in Advanced Mathematics 16, Amer. Math. Soc. and International Press, Providence and Boston, 2000.
  • Gesztesy, F. and Holden, H.: Darboux-type transformations and hyperelliptic curves. J. Reine Angew. Math. 527 (2000), 151--183.
  • Gesztesy, F. and Ratnaseelan, R.: An alternative approach to algebro-geometric solutions of the AKNS hierarchy. Rev. Math. Phys. 10 (1998), 345--391.
  • Gesztesy, F., Ratnaseelan, R. and Teschl, G.: The KdV hierarchy and associated trace formulas. In: Recent Developments in Operator Theory and Its Applications (I.,Gohberg, P. Lancaster, and P.,N. Shivakumar, editors). Operator Theory: Advances and Applications 87, 125--163. Birkhäuser, Basel, 1996.
  • Johnson, R.,S.: Camassa--Holm, Korteweg--de Vries and related models for water waves. J. Fluid Mech. 455 (2002), 63--82.
  • Marsden, J.,E., Ratiu, T.,S. and Shkoller, S.: The geometry and analysis of the averaged Euler equations and a new diffeomorphism group. Geom. Funct. Anal 10 (2000), 582--599.
  • Marsden, J.,E. and Shkoller, S.: Global well-posedness for the Lagrangian averaged Navier-Stokes (LANS-$\alpha$) equations on bounded domains. R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 359 (2001), 1449--1468.
  • Misiołek, G.: A shallow water equation as a geodesic flow on the Bott-Virasoro group. J. Geom. Phys. 24 (1998), 203--208.
  • Mumford, D.: Tata Lectures on Theta II. Progress in Mathematics 43, Birkhäuser, Boston, 1984.
  • Novikov, D.,P.: Algebraic-geometric solutions of the Harry Dym equation. Siberian Math. J. 40 (1999), 136--140.
  • Schiff, J.: Zero curvature formulations of dual hierarchies. J. Math. Phys. 37 (1996), 1928--1938.
  • Shkoller, S.: Geometry and curvature of diffeomorphism groups with $H^1$ metric and mean hydrodynamics. J. Funct. Anal. 160 (1998), 337--365.
  • Shkoller, S.: On incompressible averaged Lagrangian hydrodynamics. Preprint, arXiv:math.AP/9908109.
  • Toda, M.: Theory of Nonlinear Lattices, second edition. Solid-State Sciences 20, Springer, Berlin, 1989.
  • Xin, Z. and Zhang, P.: On the weak solutions to a shallow water equation. Comm. Pure Appl. Math. 53 (2000), 1411--1433.