Solitary and Periodic Waves in Modified Korteweg–de Vries Equation as an Inverse Problem for Coefficient Identification

This work develops numerical methods for the modified Korteweg–de Vries (mKdV) equation based on an inverse-problem formulation that identifies key solution parameters. The resulting inverse problems are treated using a variational approach that reformulates the original ill-posed formulation into an alternative well-posed minimization problem suitable for numerical computation. Numerical experiments with solitary and periodic waves demonstrate the accuracy, stability, and robustness of the proposed methods. In addition, the inverse problem framework introduced here represents a significant contribution of this study, because it provides a new computational approach for obtaining periodic solutions that existing methods cannot capture.