Abstract
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 \(\Omega \left( {x + y;t} \right) = Ce^{ - \left( {x + y} \right)A} e^{8A^3 t} B\)B, 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, 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°Q + QA = C°C and AN + NA° = BB°, where B°denotes the conjugate transposed matrix. We consider two interesting examples.
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Prepared from an English manuscript submitted by the author; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 168, No. 1, pp. 35–48, July, 2011.
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Demontis, F. Exact solutions of the modified Korteweg-de Vries equation. Theor Math Phys 168, 886–897 (2011). https://doi.org/10.1007/s11232-011-0072-4
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DOI: https://doi.org/10.1007/s11232-011-0072-4