2010/10/08 by Francesco Demontis, Demontis, Francesco
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #math-ph #math.MP
paper · pdf · doi:10.48550/arxiv.1010.1651
15 pages, 1 figure
arxiv created 2010/10/08 · openalex publication_date 2010/10/08 · arxiv updated 2010/10/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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)Ae8A3 tB, 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.