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dc.contributor.authorDeng, Shengfuen
dc.date.accessioned2014-03-14T20:13:51Zen
dc.date.available2014-03-14T20:13:51Zen
dc.date.issued2008-06-30en
dc.identifier.otheretd-07102008-135733en
dc.identifier.urihttp://hdl.handle.net/10919/28254en
dc.description.abstractThree-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) â 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) â 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.en
dc.publisherVirginia Techen
dc.relation.haspartDSF-thesis-5.pdfen
dc.rightsIn Copyrighten
dc.rights.urihttp://rightsstatements.org/vocab/InC/1.0/en
dc.subjectperiodic orbitsen
dc.subjectthree-dimensional solitary waveen
dc.subjectcenter manifoldsen
dc.subjecthomoclinic orbitsen
dc.subjectcoupled Schrödinger-KdV equationsen
dc.subjectKdV equationen
dc.subjectnormal formen
dc.titleA Spatial Dynamic Approach to Three-Dimensional Gravity-Capillary Water Wavesen
dc.typeDissertationen
dc.contributor.departmentMathematicsen
dc.description.degreePh. D.en
thesis.degree.namePh. D.en
thesis.degree.leveldoctoralen
thesis.degree.grantorVirginia Polytechnic Institute and State Universityen
thesis.degree.disciplineMathematicsen
dc.contributor.committeechairSun, Shu-Mingen
dc.contributor.committeememberRenardy, Michael J.en
dc.contributor.committeememberRussell, David L.en
dc.contributor.committeememberKim, Jong Uhnen
dc.identifier.sourceurlhttp://scholar.lib.vt.edu/theses/available/etd-07102008-135733/en
dc.date.sdate2008-07-10en
dc.date.rdate2008-07-18en
dc.date.adate2008-07-18en


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