Camassa–Holm equation

In fluid dynamics, the Camassa–Holm equation is the integrable, dimensionless and non-linear partial differential equation

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

The equation was introduced by Camassa and HolmCamassa & Holm 1993] as a bi-Hamiltonian model for waves in shallow water, and in this context the parameter $kappa$ is positive and the solitary wave solutions are smooth solitons.

Relation to waves in shallow water

: $egin{align} u_t + u u_x + p_x &= 0, \ p - p_{xx} &= 2 kappa u + u^2 + frac{1}{2} left( u_x
ight)^2, end{align}$

with "p" the (dimensionless) pressure or surface elevation. This shows that the Camassa–Holm equation is a model for shallow water waves with non-hydrostatic pressure and a water layer on a horizontal bed.

The linear dispersion characteristics of the Camassa–Holm equation are:

: $omega = 2kappa frac{k}{1+k^2},$

with "ω" the angular frequency and "k" the wavenumber. Not surprisingly, this is of similar form as the one for the Korteweg–de Vries equation, provided "κ" is unequal to zero. For "κ" equal to zero, the Camassa–Holm equation has no frequency dispersion — moreover, the linear phase speed is zero for this case. As a result, "κ" is the phase speed for the long-wave limit of "k" approaching zero, and the Camassa–Holm equation is (if "κ" is non-zero) a model for one-directional wave propagation like the Korteweg–de Vries equation.

Hamiltonian structure

Introducing the potential "m" as

: $m = u - u_{xx}, ,$

: $egin{align} m_t &= -mathcal{D}_1 frac{delta mathcal{H}_1}{delta m} & & ext{ with }& mathcal{D}_1 &= (m+kappa) frac{partial}{partial x} + frac{partial}{partial x} (m+kappa) & ext{ and } mathcal{H}_1 &= frac{1}{2} int u^2 + left(u_x
ight)^2; ext{d}x, \ m_t &= -mathcal{D}_2 frac{delta mathcal{H}_2}{delta m} & & ext{ with }& mathcal{D}_2 &= frac{partial}{partial x} + frac{partial^3}{partial x^3} & ext{ and } mathcal{H}_2 &= frac{1}{2} int u^3 + u left(u_{xx}
ight)^2 - kappa u^2; ext{d}x.end{align}$

Integrability

The Camassa–Holm equation is an integrable system. 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

: $u-u_{xx}+ kappa ,$

is positive – see Citation
last1 = Constantin
first1 = Adrian
last2 = McKean
first2 = Henry P.
year = 1999
title = A shallow water equation on the circle
journal = Commun. Pure Appl. Math.
volume = 52
issue = 8
pages = 949–982
doi = 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D
doilabel = 10.1002/(SICI)1097-0312(199908)52:8949::AID-CPA33.0.CO;2-D] and [Citation
last = Constantin
first = Adrian
year = 2001
title = On the scattering problem for the Camassa–Holm equation
journal = Proceedings of the Royal Society of London, Series A
volume = 457
pages = 953–970] for a detailed description of the spectrum associated to the isospectral problem, for the inverse spectral problem in the case of spatially periodic smooth solutions, and [Citation
last1 = Constantin
first1 = A.
last2 = Gerdjikov
first2 = V. S.
last3 = Ivanov
first3 = R. S.
year = 2006
title = Inverse scattering transform for the Camassa–Holm equation
journal = Inverse Problems
volume = 22
pages = 2197–2207
doi = 10.1088/0266-5611/22/6/017] for the inverse scattering approach in the case of smooth solutions that decay at infinity.

Exact solutions

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 [Citation
last1 = Drazin
first1 = P. G.
last2 = Johnson
first2 = R. S.
title = Solitons: an introduction
publisher = Cambridge University Press, Cambridge
year = 1989] . In the limiting case when $kappa=0$ 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 [Beals & Sattinger & Szmigielski 1999] . For the smooth solitons the soliton interactions are less elegant [Parker 2005] . 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 Constantin & Strauss 2002] – but explicit formulas are not available. Notice also that the solitary waves are orbitally stable i.e. their shape is stable under small perturbations, both for the smooth solitons and for the peakons. [ Constantin & Strauss 2000]

Wave breaking

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 [Cite book
last = Whitham
first = G. B.
author-link =
year = 1999
title = Linear and nonlinear waves
publisher = Wiley Interscience, New York–London–Sydney
year = 1974] being characterized 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 and these considerations were subsequently put on a firm mathematical basis. [Citation
last1 = Constantin
first1 = A.
last2 = Escher
first2 = J.
year = 1998
title = Wave breaking for nonlinear nonlocal shallow water equations
journal = Acta Mathematica
volume = 181
pages = 229–243
doi = 10.1007/BF02392586] It is known that the only way singularities can occur in solutions is in the form of breaking waves. [Citation
last = Constantin
first = A.
year = 2000
title = Existence of permanent and breaking waves for a shallow water equation: a geometric approach
journal = Ann. Inst. Fourier (Grenoble)
volume = 50
pages = 321–362] [Citation
last1 = Constantin
first1 = A.
last2 = Escher
first2 = J.
year = 2000
title = On the blow-up rate and the blow-up set of breaking waves for a shallow water equation
journal = Math. Z.
volume = 233
pages = 75–91
doi = 10.1007/PL00004793] 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. [Cite journal
last = McKean
first = H. P.
year = 2004
title = Breakdown of the Camassa–Holm equation
journal = Comm. Pure Appl. Math.
volume = 57
pages = 416–418
doi = 10.1002/cpa.20003] As for the continuation of solutions after wave breaking, two scenarios are possible: the conservative case [Citation
last1 = Bressan
first1 = A.
last2 = Constantin
first2 = A.
year = 2007
title = Global conservative solutions of the Camassa–Holm equation
journal = Arch. Ration. Mech. Anal.
volume = 183
pages = 215–239
doi = 10.1007/s00205-006-0010-z] and the dissipative case [Citation
last1 = Bressan
first1 = A.
last2 = Constantin
first2 = A.
year = 2007
title = Global dissipative solutions of the Camassa–Holm equation
journal = Anal. Appl.
volume = 5
pages = 1–27
doi = 10.1142/S0219530507000857] (with the first characterized by conservation of the energy, while the dissipative scenario accounts for loss of energy due to breaking).

References

*Citation
last1 = Camassa
first1 = Roberto
last2 = Holm
first2 = Darryl D.
year = 1993
title = An integrable shallow water equation with peaked solitons
journal = Phys. Rev. Lett.
volume = 71
issue = 11
pages = 1661–1664
doi = 10.1103/PhysRevLett.71.1661

*Citation
last1 = Beals
first1 = R.
last2 = Sattinger
first2 = D.
last3 = Szmigielski
first3 = J.
year = 1999
title = Multi-peakons and a theorem of Stieltjes
journal = Inverse Problems
volume = 15
pages = L1–L4
doi = 10.1088/0266-5611/15/1/001

*Cite journal
last = Parker
first = A.
year = 2005
title = On the Camassa–Holm equation and a direct method of solution. N-soliton solutions.
journal = Proceedings Royal Society of London, Series A
volume = 461
pages = 3893–3911
doi = 10.1098/rspa.2005.1537

*Citation
last1 = Constantin
first1 = A.
last2 = Strauss
first2 = W. A.
year = 2002
title = Stability of the Camassa–Holm solitons
journal = J. Nonlinear Science
volume = 12
pages = 415–422
doi = 10.1007/s00332-002-0517-x

*Citation
last1 = Constantin
first1 = A.
last2 = Strauss
first2 = W. A.
year = 2000
title = Stability of peakons
journal = Comm. Pure Appl. Math.
volume = 53
pages = 603–610
doi = 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L
doilabel = 10.1002/(SICI)1097-0312(200005)53:5603::AID-CPA33.0.CO;2-L

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