Initial boundary value problem for the mKdV equation on a finite interval. (Q1774069)

The authors consider the modified Korteweg-de Vries (mKdV) equation \[ q_t-q_{xxx}+6\lambda q^2 q_x=0,\quad \lambda=\pm 1 \] on a finite interval. To show the existence of solutions with prescribed initial and boundary data, they implement the method based on the simultaneous spectral analysis of the two eigenvalue equations of the associated Lax pair as outlined by \textit{A. S. Fokas} [Proc. Roy. Soc. London, Ser. A, 453, No. 1962, 1411--1443 (1997; Zbl 0876.35102)].