Existence of global solutions for the modified Camassa-Holm equation with a nonzero background

Transient asymptotics of the modified Camassa–Holm equation

“…The global well-posedness of the Cauchy problem for the mCH equation under the above assumption has recently been established in [47]. Indeed, it was also shown therein that 𝑚 − 1 ∈ 𝐶([0, +∞), 𝐻 2,1 (ℝ)) ∩ 𝐻 1,2…”

“…Suppose the initial data satisfy Assumption 2.1, it has been shown in [47] that the scattering data Proposition 3.2. Let 𝑀 (1) (𝑧; 𝑦, 𝑡) be the unique solution of RH problem 3.1, we have 𝑀 (1) (0; 𝑦, 𝑡) =…”

“…Similar to the CH equation, the mCH equation possesses a bi-Hamiltonian structure [39] and the wave-breaking, peakon solutions [28]; see also [31] for the quasi-periodic algebro-geometric solutions. The Bäcklund transformation and the related nonlinear superposition formulae were presented in [44] and the existence of global solutions were analyzed in [47].…”