Title page for ETD etd-07102008-135733

Type of Document Dissertation
Author Deng, Shengfu
Author's Email Address sfdeng@vt.edu
URN etd-07102008-135733
Title A Spatial Dynamic Approach to Three-Dimensional Gravity-Capillary Water Waves
Degree PhD
Department Mathematics
Advisory Committee
Advisor Name Title
Sun, Shu-Ming Committee Chair
Kim, Jong Uhn Committee Member
Renardy, Michael J. Committee Member
Russell, David L. Committee Member
  • periodic orbits
  • three-dimensional solitary wave
  • center manifolds
  • homoclinic orbits
  • coupled Schrödinger-KdV equations
  • KdV equation
  • normal form
Date of Defense 2008-06-30
Availability unrestricted
Three-dimensional gravity-capillary steady waves on water of finite-depth, which are uniformly translating in a horizontal propagation direction and periodic in a transverse direction, are considered. The exact Euler equations are formulated as a spatial dynamic system in which the variable used for the propagating direction is the time-like variable. The existence of the solutions of the system is determined by two non-dimensional constants: the Bond number b and λ (the inverse of the square of the Froude number). The property of Sobolev spaces and the spectral analysis show that the spectrum of the linear part consists of isolated eigenvalues of finite algebraic multiplicity and the number of purely imaginary eigenvalues are finite. The distribution of eigenvalues is described by b and λ.

Assume that C1 is the curve in (b,λ)-plane on which the first two eigenvalues for three-dimensional waves collide at the imaginary axis, and that the intersection point of the curve C1 with the line λ=1 is (b0,1) where b0>0. Two cases (b0,1) and (b,λ) ∈ C1 where 0< b< b0 are investigated. A center-manifold reduction technique and a normal form analysis are applied to show that for each case the dynamical system can be reduced to a system of ordinary differential equations with finite dimensions. The dominant system for the case (b0,1) is coupled Schrödinger-KdV equations while it is a Schrödinger equation for another case (b,λ) ∈ C1. Then, from the existence of the homoclinic orbit connecting to the two-dimensional periodic solution (called generalized solitary wave) for the dominant system, it is obtained that such generalized solitary wave solution persists for the original system by using the perturbation method and adjusting some appropriate constants.

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