Camassa–Holm equation

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Interaction of two peakons — which are sharp-crested soliton solutions to the Camassa–Holm equation. The wave profile (solid curve) is formed by the simple linear addition of two peakons (dashed curves):
u = m_1 \, e^{-|x-x_1|} + m_2 \, e^{-|x-x_2|}.
The evolution of the individual peakon positions x_1(t) and x_2(t), as well as the evolution of the peakon amplitudes m_1(t) and m_2(t), is however less trivial: this is determined in a non-linear fashion by the interaction.
Traveling wave solution of Camassa Holm eq

In fluid dynamics, the Camassa–Holm equation is the integrable, dimensionless and 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}. \,

The equation was introduced by Camassa and Holm[1] as a bi-Hamiltonian model for waves in shallow water, and in this context the parameter κ is positive and the solitary wave solutions are smooth solitons.

In the special case that κ is equal to zero, the Camassa–Holm equation has peakon solutions: solitons with a sharp peak, so with a discontinuity at the peak in the wave slope.

Relation to waves in shallow water[edit]

The Camassa–Holm equation can be written as the system of equations:[2]


\begin{align}
  u_t + u u_x + p_x &= 0, \\
  p - p_{xx} &= 2 \kappa u + u^2 + \frac{1}{2} \left( u_x \right)^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 = 2\kappa \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 non-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[edit]

Introducing the momentum m as

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

then two compatible Hamiltonian descriptions of the Camassa–Holm equation are:[3]


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

Integrability[edit]

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 the momentum


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

is positive — see [4] and [5] for a detailed description of the spectrum associated to the isospectral problem,[4] for the inverse spectral problem in the case of spatially periodic smooth solutions, and [6] for the inverse scattering approach in the case of smooth solutions that decay at infinity.

Exact solutions[edit]

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. [7] 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.[8] For the smooth solitons the soliton interactions are less elegant.[9] 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[10] — 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[10] and for the peakons.[11]

Camassa Holm equation traveling wave sech plot5 
Camassa Holm equation traveling wave sech plot4 
Camassa Holm equation traveling wave sech plot6 
Camassa Holm equation traveling wave exp plot2 
Camassa Holm equation traveling wave exp plot3 
Camassa Holm equation traveling wave exp plot5 
Camassa Holm equation traveling wave sec plot3 
Camassa Holm equation traveling wave sec plot5 

Wave breaking[edit]

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[12] 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[1] and these considerations were subsequently put on a firm mathematical basis.[13] It is known that the only way singularities can occur in solutions is in the form of breaking waves.[14] 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.[15] As for the continuation of solutions after wave breaking, two scenarios are possible: the conservative case[16] and the dissipative case[17] (with the first characterized by conservation of the energy, while the dissipative scenario accounts for loss of energy due to breaking).

Long-time asymptotics[edit]

It can be shown that for sufficiently fast decaying smooth initial conditions with positive momentum splits into a finite number and solitons plus a decaying dispersive part. More precisely, one can show the following for \kappa>0:[18] Abbreviate c =x / (\kappa t). In the soliton region c>2 the solutions splits into a finite linear combination solitons. In the region 0<c<2 the solution is asymptotically given by a modulated sine function whose amplitude decays like t^{-1/2}. In the region -1/4<c<0 the solution is asymptotically given by a sum of two modulated sine function as in the previous case. In the region c<-1/4 the solution decays rapidly. In the case \kappa=0 the solution splits into an infinite linear combination of peakons[19] (as previously conjectured[20]).

Peaked waves in finite water depth[edit]

In 2003, a unified wave model (UWM) for progressive gravity waves in finite water depth was proposed by Liao. Based on the symmetry and the exact wave equations, the UWM admits not only all traditional smooth periodic/solitary waves but also the peaked solitary waves including the famous peaked solitary waves of Camassa–Holm equation mentioned above. Thus, the UWM unifies the smooth and peaked waves in finite water depth. In other words, the peaked solitary waves are consistent with the traditional, smooth ones, and thus are as acceptable as the smooth ones.

It is found that the peaked solitary waves in finite water depth have some unusual characteristics. First of all, it has a peaked wave elevation with a discontinuous vertical velocity v at crest. Secondly, unlike the smooth waves whose horizontal velocity u decays exponentially from free surface to the bottom, the horizontal velocity u of the peaked solitary waves always increases from free surface to the bottom. Especially, different from the smooth waves whose phase speed is dependent upon wave height, the phase speed of the peaked solitary waves in finite water depth have nothing to do with the wave height! In other words, the peaked solitary waves in finite water depth are non-dispersive.

The above usual characteristics of the peaked solitary waves in finite water depth are quite different from those of the traditional, smooth waves, and thus might challenge some traditional viewpoints. Even so, they could enrich and deepen our understandings about the peaked solitary waves and the Camassa–Holm equation.

See also[edit]

Notes[edit]

References[edit]

Further reading[edit]