Camassa-Holm equation

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In the theory of integrable systems, the Camassa-Holm equation is the integrable non-linear partial differential equation

u_t + 2\kappa u_x - u_{xxt} + 3 u u_x = 2 u_x u_{xx} + u u_{xxx}. \,

Integrability means that there is a change of variables (action-angle variables) such that the evolution equation in the new variables is equivalent to a linear flow at constant speed. This change of variables is achieved by studying an associated isospectral/scattering problem, and is reminiscent of the fact that integrable classical Hamiltonian systems are equivalent to linear flows at constant speed on tori. The Camassa-Holm equation is integrable provided that

m = u-u_{xx}+ \kappa \,

is positive – see [1] and [2] for a detailed description of the spectrum associated to the isospectral problem, [1] for the inverse spectral problem in the case of spatially periodic smooth solutions, and [3] for the inverse scattering approach in the case of smooth solutions that decay at infinity.

The equation was introduced by Camassa and Holm[4] as a model for waves in shallow water, and in this context the parameter κ is positive and the solitary wave solutions are smooth solitons. Traveling waves are solutions of the form

u(t,x)=f(x-ct) \,

representing waves of permanent shape f that propagate at constant speed c. These waves are called solitary waves if they are localized disturbances, that is, if the wave profile f decays at infinity. If the solitary waves retain their shape and speed after interacting with other waves of the same type, we say that the solitary waves are solitons. There is a close connection between integrability and solitons [5]. In the limiting case when κ = 0 the solitons become peaked (shaped like the graph of the function f(x) = e − | x | ), and they are then called peakons. It is possible to provide explicit formulas for the peakon interactions, visualizing thus the fact that they are solitons [6]. For the smooth solitons the soliton interactions are less elegant [7]. This is due in part to the fact that, unlike the peakons, the smooth solitons are relatively easy to describe qualitatively – they are smooth, decaying exponentially fast at infinity, symmetric with respect to the crest, and with two inflection points [8] – but explicit formulas are not available. Notice also that the solitary waves are orbitally stable i.e. their shape is stable under small perturbations cf. [8] for the smooth solitons and [9] for the peakons.

The Camassa-Holm equation models breaking waves: a smooth initial profile with sufficient decay at infinity develops into either a wave that exists for all times or into a breaking wave (wave breaking [10] being charracterized by the fact that the solution remains bounded but its slope becomes unbounded in finite time). The fact that the equations admits solutions of this type was discovered by Camassa and Holm [4] and these considerations were subsequently put on a firm mathematical basis by Constantin and Escher [11]. It is known that the only way singularities can occur in solutions is in the form of breaking waves [12], [13]. Moreover, from the knowledge of a smooth initial profile it is possible to predict (via a necessary and sufficient condition) whether wave breaking occurs or not [14]. As for the continuation of solutions after wave breaking, two scenarios are possible: the conservative case [15] and the dissipative case [16] (with the first characterized by conservation of the energy, while the dissipative scenario accounts for loss of energy due to breaking).

[edit] Notes

  1. ^ a b Constantin & McKean 1999
  2. ^ Constantin 2001
  3. ^ Constantin & Gerdjikov & Ivanov 2006
  4. ^ a b Camassa & Holm 1993
  5. ^ Drazin & Johnson 1989
  6. ^ Beals & Sattinger & Szmigielski 1999
  7. ^ Parker 2005
  8. ^ a b Constantin & Strauss 2002
  9. ^ Constantin & Strauss 2000
  10. ^ Whitham 1974
  11. ^ Constantin & Escher 1998
  12. ^ Constantin 2000
  13. ^ Constantin & Escher 2000
  14. ^ McKean 2004
  15. ^ Bressan & Constantin 2007
  16. ^ Bressan & Constantin 2007

[edit] References

  • Constantin, Adrian & McKean, Henry P. (1999), “A shallow water equation on the circle”, Commun. Pure Appl. Math. 52 (8): 949–982, DOI 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D 
  • Constantin, Adrian (2001), “On the scattering problem for the Camassa-Holm equation”, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 457: 953–970 
  • Constantin, A.; Gerdjikov, V. S. & Ivanov, R. S. (2006), “Inverse scattering transform for the Camassa-Holm equation”, Inverse Problems 22: 2197–2207 
  • Drazin, P. G. & Johnson, R. S. (1989), Solitons: an introduction, Cambridge University Press, Cambridge 
  • Beals, R.; Sattinger, D. & Szmigielski, J. (1999), “Multi-peakons and a theorem of Stieltjes”, Inverse Problems 15: L1–L4 
  • Parker, A. (2005), “On the Camassa-Holm equation and a direct method of solution. N-soliton solutions.”, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461: 3893–3911 
  • Constantin, A. & Strauss, W. A. (2002), “Stability of the Camassa-Holm solitons”, J. Nonlinear Science 12: 415–422 
  • Constantin, A. & Strauss, W. A. (2000), “Stability of peakons”, Comm. Pure Appl. Math. 53: 603–610 
  • Constantin, A. & Escher, J. (1998), “Wave breaking for nonlinear nonlocal shallow water equations”, Acta Mathematica 181: 229–243 
  • Constantin, A. & Escher, J. (2000), “On the blow-up rate and the blow-up set of breaking waves for a shallow water equation”, Math. Z. 233: 75–91 
  • Constantin, A. (2000), “Existence of permanent and breaking waves for a shallow water equation: a geometric approach”, Ann. Inst. Fourier (Grenoble) 50: 321–362 
  • McKean, H. P. (2004), “Breakdown of the Camassa-Holm equation”, Comm. Pure Appl. Math. 57: 416–418 
  • Whitham, G. B. (1974), Linear and nonlinear waves, Wiley Interscience, New York–London–Sydney 
  • Bressan, A. & Constantin, A. (2007), “Global conservative solutions of the Camassa-Holm equation”, Arch. Ration. Mech. Anal. 183: 215–239 
  • Bressan, A. & Constantin, A. (2007), “Global dissipative solutions of the Camassa-Holm equation”, Anal. Appl. 5: 1–27