Genesis and extinction of solitons arising from individual flows of the Camassa-Holm hierarchy: The rise of a novel Darboux-like transform .

摘要

The present work offers a detailed account, of the large time development of the velocity profile v run by a single “individual” Hamiltonian flow of the Camassa-Holm (CH) hierarchy, the Hamiltonian employed being the invariant H = 1/$l$, where $l$ is any of the bound states of the associated spectral problem: (¼ − D2)(f) = $l$mf, with “mass” potential m ≡ v − v″. The flow may be expressed as in ∂ m/∂t = [mD + Dm]( f2) = 1/(2$l$)D(1 − D2)( f2), or more simply, as ∂v/∂ t = 1/(2$l$)D(f2). Unlike the formation of the soliton train that is produced by Korteweg-de Vries (KdV) ∂ V/∂t = 3V∂V/∂ X − ½∂3V/∂ X3, which accounts, except for the reflectionless potential V, only for the part of the total energy ascribed to the bound states of the associated spectral problem (−D 2 + V)(F) = $l$F, the deficiency being carried by the evanescent radiation corresponding to the continuous spectrum; for summable m , CH has only bound states $ln$, n ∈ $Z$ − {lcub}0{rcub}, each of which characterizes the speed=amplitude of the associated individual soliton $Sn$(t, x) ≡ 1/(2$ln$). They embody respectively an energy ½ ∫[($S\prime n$)2 + $S2n$] = 1/(4$l2n$), and all these individual pieces add up to the whole: H_CH ≡ ½ ∫ mG[m] = ∑1/(4$l2n$) where G ≡ (1 − D2) −1, so here nothing is lost. And indeed, the present investigation confirms this:; Let m be summable and odd, having the signature of x, and consider the individual flow based upon H = 1/$l$ with $l$ > 0. With the help of a private “Lagrangian” scale determined by x¯· = −f 2(t, x¯) and x¯(0, x) = x; the updated velocity profile ($etXH$v(0, ·)) (x¯) ≡ v(t, x¯(t, x)) is found to shape itself like the soliton $Sl$(t, x) = 1/(2 esc