1. Introduction & Purposes

This program introduces undergraduate researchers to analytical and computational methods for studying traveling wave solutions of evolutional partial differential equations and their stability. A central theme is the distinction between absolute and convective behavior of perturbations, both for homogeneous states and for traveling wave solutions viewed in a comoving frame. The project combines complex analysis (via the Fokas method), numerical simulation, and spectral analysis, guiding students from foundational concepts to focused research projects.

2. Outline & Descriptions

The program will run from July 20 to July 30, 2026. During this period, students will participate in lectures and research projects under the supervision of Profs. Hsin-Yuan Huang, Te-Sheng Lin, and Chang-Hong Wu from National Yang Ming Chiao Tung University, together with three graduate student mentors. Students are encouraged to have the relevant prerequisite knowledge. The detailed description of the program is provided below.

l  Lectures

l  The lectures will cover the following three topics

○         Spectral Representations and the Fokas Method

■         Lecture 1: Fokas method for linear PDEs with various boundary conditions

■         Lecture 2. Fokas method for systems of linear PDEs

■         Lecture 3. Absolute and convective instability for evolutionary PDEs

■         Lecture 4. Wrap-up and discussion

○         Numerical Methods and Visualization of Instability

○         Analysis of Traveling Wave Solutions

l  Independent Research projects

Students will work in small groups on focused research projects under faculty supervision. Possible project directions include:

•           Classification of absolute versus convective instability for selected parameter regimes

•           Comparison of numerical simulations with spectral predictions

•           Study of boundary effects using Fokas-type representations

•           Investigation of the parametric dependence of wave speed and instability thresholds

•           Global/local stability analysis to traveling waves in selected models

l  References

○      M. Ablowitz and A. S Fokas (2003), "Complex variables: introduction and applications". Cambridge University Press.

○      B. Deconinck, T. Trogdon, and V. Vasan, "The method of Fokas for solving linear partial differential equations", *SIAM Review* 56 (2014) pp.159-186.

○      P. C. Fife, J. B. McLeod, The approach of solutions of nonlinear diffusion equationstotravellingfrontsolutions,Arch.Ration.Mech.Anal.65(4)(1977) 335–361

○      A.S.FokasandD.T.Papageorgiou,"Absoluteandconvectiveinstabilityfor evolutionPDEsonthehalf-line",*Studies in Applied Mathematics*, 114 (2005)

pp.95–114.

○      T.KapitulaandK.Promislow(2013),"SpectralandDynamicalStabilityof Nonlinear Waves", Springer.

○      H.Kokubu,Homoclinicandheteroclinicbifurcationsofvectorfields,JapanJ. Appl. Math., 5 (1988), pp. 455–501.

3. Registration

https://forms.gle/kvwZdeBEU8DLGt4d9