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Showing 1 to 11 of 11 for “"KdV equation"”.
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Exact solutions and conservation laws of two-wave mode Kadomtsev-Petviashvili equation and a nonintegrable nonlinear equation
… we study three nonlinear partial differential equations namely the Korteweg-de Vries (KdV) equation as an illustrative example, nonintegrable nonlinear equation and two-wave mode Kadomtsev-Petviashvili (TKP) equation. We employ the multiplier method to find conservation laws. We also use …
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Whitham Equations, Dispersionless KP Theory and Seiberg-Witten Variables
We discuss averaging methods to get Whitham equations for the KP hierarchy and use period integrals, which correspond to SW variables, in the Whitham equations. Then we study applications of the Whitham theory and formulas involving branch points of some Riemann Surfaces associated with the KdV …
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A study of certain multi-dimensional partial differential equations using Lie symmetry analysis
… nonlinear multi-dimensional partial differential equations which are mathematical models of various physical phenomena of the real world. Closed-form solutions and conservation laws are obtained for such equations using various methods. The multi-dimensional partial differential equations that are …
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Integrable Evolution Partial Differential Equations: Multidimensions and Dirichlet-to-Neumann Maps
… thesis is about integrable Partial Differential Equations (PDEs). Here "integrable" means that the given PDE or system of PDEs can be written as the compatibility condition of a set of linear equations, the so-called Lax pair, and can be solved via a method related to the Inverse Scattering …
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Self-similar singularity formation and wellposedness theory for compressible fluids and dispersive PDE
… formation and local wellposedness of fluid equations and dispersive PDE. Regarding singularity formation, we construct radially symmetric smooth selfsimilar profiles for the compressible Euler equations which exhibit an implosion type singularity in finite time. This will be the first part …
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Nonsemilinear one-dimensional PDEs: analysis of PT deformed models and numerical study of compactons
… the complex PT-symmetric deformations of the KdV equation and of the inviscid Burgers equation, respectively. The second part of the thesis, comprising Chapters 3 and 4, contains a review and original numerical studies on the properties of certain quasilinear dispersive PDEs in one dimension …
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Discrete Fourier restriction phenomenon associated with some periodic dispersive equations
… phenomenon associated with Schrodinger equations. We study the size of the Fourier transform of a periodic function on a truncated discrete paraboloid. We develop two ways to tackle the problem, and the second one recovers Bourgain's level set result on Strichartz estimates associated …
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Uncertainty quantification and prediction for non-autonomous linear and nonlinear systems
… schemes and dynamically orthogonal (DO) field equations. In this thesis, we focus our attention on DO and various PC schemes for quantifying and predicting uncertainty in systems with external stochastic forcing. We develop and implement these schemes in a generic stochastic solver for a class …
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Soliton solutions of noncommutative integrable systems
… either the hierarchy or Lax pair generating the equation. Both Chapters 1 and 2 are entirely made up of background material and contain no new material. Furthermore, these chapters are concerned with commutative equations. Chapter 1 outlines some of the basic concepts of integrable systems …
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Analysis and Implementation of High-Order Compact Finite Difference Schemes
… boundary value problems modeled by differential equations. These schemes generally require smaller stencils than the traditional explicit finite difference counterparts. To avoid numerical instabilities at and near boundaries and in regions of mesh non-uniformity, a numerical filtering technique …
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A Spatial Dynamic Approach to Three-Dimensional Gravity-Capillary Water Waves
… 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 …