Initial–boundary problems for the vector modified Korteweg–de Vries equation via Fokas unified transform method

Abstract

Initial–boundary problems for the vector modified Korteweg–de Vries equation on the half-line are investigated by Fokas unified transform method. Even though additional technical complications arise in the multi-component case compared with scalar ones, it is shown that the solution q(x,t) can be expressed in terms of the solution of a matrix Riemann–Hilbert problem. The Riemann–Hilbert problem involves a jump matrix, uniquely defined in terms of four matrix spectral functions denoted by {a(λ),b(λ),A(λ),B(λ)} that depend on the initial data and all boundary values, respectively. A key role is played by the so-called global relation which involves the known and unknown boundary values. By analyzing the global relation, an effective characterization of the latter two spectral functions is presented.