Soliton resolution for the modified short pulse equation with the weighted Sobolev initial data

In this paper, the Cauchy problem of the modified short pulse (mSP) equation with initial conditions in weighted Sobolev space is studied by using (partial derivative) over bar -steepest descent method. Based on the spectral analysis of Lax pair and the transformations of field variables and independent variables, a basic Riemann-Hilbert problem (RHP) is established, and the solution of Cauchy problem for the mSP equation is transformed into the corresponding RHP. The asymptotic expansion of the solution of mSP equation is derived in a fixed space-time cone C((x) over tilde (1), (x) over tilde (2),v(1),v(2)) = {((x) over tilde ,t) is an element of R- x R+|(x) over tilde = (x) over tilde (0)+vt, with (x) over tilde (0) is an element of [(x) over tilde (1),(x) over tilde (2)],v is an element of [v(1),v(2)]}. According to this, the soliton resolution conjecture of the mSP equation is given, in which the leading term is characterized by an N(I)-soliton on the discrete spectrum, the second term comes from the soliton-radiation interactions on the continuum spectrum, and the error term O(t(-3)/4) is generated by the corresponding (partial derivative) over bar -problem.