Exact solutions of the modified Korteweg-de Vries equation

We use the inverse scattering method to obtain a formula for certain exact solutions of the modified Korteweg-de Vries (mKdV) equation. Using matrix exponentials, we write the kernel of the relevant Marchenko integral equation as Omega(x_y;t) = Ce(-(x+y)A)e(8A3t)B, where the real matrix triplet (A,B,C) consists of a constant pxp matrix A with eigenvalues having positive real parts, a constant px1 matrix B, and a constant 1x p matrix C for a positive integer p. Using separation of variables, we explicitly solve the Marchenko integral equation, yielding exact solutions of the mKdV equation. These solutions are constructed in terms of the unique solution P of the Sylvester equation AP + PA = BC or in terms of the unique solutions Q and N of the Lyapunov equations A(A degrees)Q + QA = C(A degrees)C and AN + NA(A degrees) = BB(A degrees), where B(A degrees)denotes the conjugate transposed matrix. We consider two interesting examples.