In this paper, we address the existence of global solutions to the Cauchy
problem of the modified Camassa-Holm (mCH) equation, which is known as a model
for the unidirectional propagation of shallow water waves. Based on the
spectral analysis of the Lax pair, we apply the inverse scattering transform to
rigorously analyze the mCH equation with zero background. By connecting the
Cauchy problem to the Riemann-Hilbert (RH) problem, we establish a bijective
map between potential and reflection coefficients within the L2-Sobolev
space framework. Utilizing a reconstruction formula and estimates on the
time-dependent RH problem, we obtain a unique global solution to the Cauchy
problem for the mCH equation.Comment: 29 page