Exact solutions to the modified Korteweg-de Vries equation

A formula for certain exact solutions to the modified Korteweg-de Vries (mKdV) equation is obtained via the inverse scattering transform method. The kernel of the relevant Marchenko integral equation is written with the help of matrix exponentials as Ω(x + y; t) = Ce−(x+y)Ae8A 3tB, where the real matrix triplet (A,B,C) consists of a constant p × p matrix A with eigenvalues having positive real parts, a constant p × 1 matrix B, and a constant 1 × p matrix C for a positive integer p. Using separation of variables, the Marchenko integral equation is explicitly solved yielding exact solutions to the mKdV equation. These solutions are constructed in terms of the unique solution P to the Sylvester equation AP +PA = BC or in terms of the unique solutions Q and N to the respective Lyapunov equations A†Q+QA = C†C and AN +NA † = BB†, where the † denotes the matrix conjugate transpose. Two interesting examples are provided. 1

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