n this paper, we study the Cauchy problem of the following complex cubic Camassa-Holm (ccCH) equation mt=bux+12[m(|u|2−|ux|2)]x−12m(uu¯x−uxu¯),m=u−uxx,
where b is an arbitrary real constant. %By applying ∂¯-steepest descent method, l Long-time asymptotics of the equation is obtained through the ∂¯-steepest descent method. Firstly, based on the spectral analysis of the Lax pair and scattering matrix, the solution of the equation is able to be constructed %can be expressed by %the solution of via solving the corresponding Riemann-Hilbert problem (RHP). Then, we present %obtain different long time asymptotic expansions of the solution u(y,t) in different space-time solitonic regions of ξ=y/t. The half-plane (y,t):−∞0 is divided into four asymptotic regions: ξ∈(−∞,−1), ξ∈(−1,0), ξ∈(0,18) and ξ∈(18,+∞). When ξ falls in (−∞,−1)∪(18,+∞), no stationary phase point of the phase function θ(z) exists on the jump profile in the space-time region. In this case, corresponding asymptotic approximations can be characterized with an N(Λ)-solitons with diverse residual error order O(t−1+2ε). %While There are four stationary phase points and eight stationary phase points on the jump curve as ξ∈(−1,0) and ξ∈(0,18), respectively. The corresponding asymptotic form is accompanied by a residual error order O(t−34).
Zhang, Hongyi, Yufeng Zhang, and Zhijun Qiao. "Long-time asymptotics of a complex cubic Camassa-Holm equation." arXiv preprint arXiv:2211.16814 (2022).