Camassa–Holm Equation - Exact Solutions

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. 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. For the smooth solitons the soliton interactions are less elegant. 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 — 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.

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