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Open Access Publications from the University of California

A Local Construction of the Smith Normal Form of a Matrix Polynomial, and Time-periodic Gravity-driven Water Waves

  • Author(s): Yu, Jia
  • Advisor(s): Wilkening, Jon A
  • et al.

This dissertation consists of two separate chapters. In the first chapter, we present an algorithm for computing a Smith normal form with multipliers of a regular matrix polynomial over a field. This algorithm differs from previous ones in that it computes a local Smith form for each irreducible factor in the determinant separately and combines them into a global Smith form, whereas other algorithms apply a sequence of unimodular operations to the original matrix row by row (or column by column) to obtain the Smith normal form. The performance of the algorithm in exact arithmetic is reported for several test cases.

The second chapter is devoted to a numerical method for computing nontrivial time-periodic, gravity-driven water waves with or without surface tension. This method is essentially a shooting method formulated as a minimization problem. The objective function depends on the initial conditions and the proposed period, and measures deviation from time-periodicity. We adapt an adjoint-based optimal control method to rapidly compute the gradient of the functional. The main technical challenge involves handling the nonlocal Dirichlet to Neumann operator of the water wave equations in the adjoint formulation. Several families of traveling waves and symmetric breathers are simulated. In the latter case, we observe disconnections in the bifurcation curves due to nonlinear resonances at critical bifurcation parameters.

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