Applied Mathematics

Computation and continuation of traveling waves in phase field models: The computation of certain traveling waves has been shown to correspond to that of heteroclinic connections between equilibrium points of a dynamical system (see [10], [17]). The problem of computation and continuation of connecting orbits between equilibrium points and periodic orbits can be fully understood from [3],[4], and [9]. In particular, in [9] the continuation of invariant subspaces algorithm from [8] is adapted into a smooth definition of the projection boundary conditions, and a general algorithm is proposed for the computation and continuation of connecting orbits. Phase field models are continuum descriptions of phase transitions from an initial to a final state, which in general correspond to certain initial and final values of a problem parameter. The existence of these transitions has been studied in [2] and [13]. In particular, in [13] a rapid solidification process is analyzed by computing traveling waves, and some numerical examples are provided. The author proposes to study the problem of transition between different phases, say from liquid to solid, from the point of view of connecting orbits between equilibrium points. Following [4] and [20], the first part of the project will be mathematical modeling to arrive to a well-posed boundary value problem. A theoretical study of existence, uniqueness and stability of solutions will then be followed by the numerical analysis and computation of the connecting orbits. We plan to show that these orbits are in fact the traveling waves we are looking for. Several numerical examples will illustrate the effectiveness of the approach. This project will be a challenging and at the same time a reasonable and exciting experience to REU participants, given the well-defined structure of the problem and the existing software and numerical codes available [9], [10], and [11] (including MATLAB) at the host institution. The author also proposes other levels of completion. The first one would start with the computed solutions above to perform continuation of these solutions as certain parameters vary, which would be done using the approaches proposed in [8] and [9]. This would lead to the computation of branches of connecting orbits in the form of traveling wave fronts, a topic with wide applications on its own. The second one would consider the generalization of the connections in the original problem to those between an equilibrium point and a periodic orbit, and to connections between periodic orbits. In this new context, some other applications arise, such as the computation of solitary waves in reduced water-wave problems [5] and space mission designs [15][15]. At these new levels, more advanced theoretical concepts (monodromy matrices, Floquet theory, etc.), and other numerical techniques (e.g. numerical computation of smooth Schur factorizations) will be needed. This would be truly challenging problems for the most ambitious and talented students, and for the continuation of REU projects in the next years.