Variational Method for Inverse Problems in Identifying Solitary-Wave Solutions

Abstract Inverse and ill-posed problems are among the most active areas of research in mathematics, science, and engineering. An inverse problem is characterized by knowing the outcomes or consequences but not the underlying causes. In the context of solitary wave equations, numerical algorithms face a significant challenge because the solitary wave arises from a bifurcation, with the problem always possessing a trivial solution. This trivial solution acts as a strong attractor, making it essential to take special measures to avoid it when using iterative methods. This work introduces a novel approach to circumvent bifurcation by reformulating the solitary wave problem as an inverse problem.