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 $\Omega(x+y;t)=Ce^{-(x+y)A}e^{8A^3 t}B$, where the real matrix triplet $(A,B,C)$ consists of a constant $p{\times}p$ matrix $A$ with eigenvalues having positive real parts, a constant $p\times1$ matrix $B$, and a constant $1\times p$ 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 $A^\dag Q+QA=C^\dag C$ and $AN+NA^\dag=BB^\dag$, where $B^\dag$ denotes the conjugate transposed matrix. We consider two interesting examples.