published_or_final_version / Physics / Doctoral / Doctor of Philosophy

by Zheng Yu-kun. / Thesis (Ph.D.)--Chinese University of Hong Kong, 1991. / Includes bibliographical references. / Preface --- p.1 / Chapter Chapter 1. --- Gauge Transformation and the Higher Order Korteweg-de Vries Equations --- p.6 / Chapter 1. --- Higher order KdV equations --- p.6 / Chapter 2. --- η2-dependent higher order mKdV equations --- p.9 / Chapter 3. --- η2-dependent Miura transformation and Backlund transformation --- p.13 / Chapter 4. --- Gauge transformation of the wave function --- p.15 / Chapter 5. --- Backlund transformation for the η2 -dependent higher order mKdV equation --- p.24 / Chapter 6. --- Applications --- p.25 / Chapter 7. --- References --- p.30 / Chapter Chapter 2. --- Solutions of a Nonisospectral and Variable Coefficient Korteweg-de Vries Equation --- p.31 / Chapter 1. --- Introduction --- p.31 / Chapter 2. --- Nonisospectral variable coefficient KdV-type equations --- p.32 / Chapter 3. --- Invariance of LP under the Crum transformation --- p.34 / Chapter 4. --- Backlund transformation for the h-t-KdV equation --- p.35 / Chapter 5. --- Solutions --- p.39 / Chapter 6. --- References --- p.43 / Chapter Chapter 3. --- Nonisospectral Variable Coefficient Higher Order Korteweg-de Vries Equations --- p.45 / Chapter 1. --- Introduction --- p.45 / Chapter 2. --- Nonisospectral t-ho-KdV equations --- p.47 / Chapter 3. --- Nonisospectral η2 dependent variable coefficient higher order modified KdV equation --- p.50 / Chapter 4. --- Backlund transformation and gauge transformation --- p.57 / Chapter 5. --- Example. Solutions of second order ni-t-KdV equation and its corresponding ni-t-η2-mKdV equation --- p.61 / Chapter 6. --- References --- p.66 / Chapter Chapter 4. --- Gauge and Backlund Transformations for the Variable Coefficient Higher-Order Modified Korteweg-de Vries Equation --- p.67 / Chapter 1. --- Introduction --- p.67 / Chapter 2. --- The t-ho-mKdV equation --- p.68 / Chapter 3. --- Some results about the t-ho-KdV equation --- p.74 / Chapter 4. --- A Backlund transformation for the t-ho-mKdV equation --- p.76 / Chapter 5. --- Gauge transformat ion and the Backlund transformation --- p.78 / Chapter 6. --- References --- p.85 / Chapter Chapter 5. --- Gauge and Backlund Transformat ions for the Generalized Sine-Gordon Equation and Its η Dependent Modified Equation --- p.86 / Chapter 1. --- Introduction --- p.86 / Chapter 2. --- Generalized Sine-Gordon equation --- p.87 / Chapter 3. --- Backlund transformation for the GSGE --- p.92 / Chapter 4. --- Gauge transformations for AKNS systems --- p.98 / Chapter 5. --- η dependent modified GSGE and its Backlund transformation --- p.102 / Chapter 6. --- Summary and example --- p.105 / Chapter 7. --- References --- p.110 / Chapter Chapter 6. --- Backlund Transformation for the Nonisospectral and Variable Coefficient Nonlinear Schrodinger Equation --- p.111 / Chapter 1. --- Introduction --- p.111 / Chapter 2. --- A generalized NLSE --- p.112 / Chapter 3. --- Γ Riccati equation system --- p.114 / Chapter 4. --- Invariance of the Γ-system --- p.116 / Chapter 5. --- Lax pair corresponding to the GNLSE --- p.119 / Chapter 6. --- BT´ةs for the Γ evolution equation and the GNLSE --- p.121 / Chapter 7. --- References --- p.126 / Chapter Chapter 7. --- Backlund Transformations for the Caudrey-Dodd-Gibbon-Sawada-Kotera Equation and Its λ-Modified Equation --- p.127 / Chapter 1. --- Introduction --- p.127 / Chapter 2. --- The CDGSKE and the λ-mCDGSKE --- p.128 / Chapter 3. --- The general solution for the scattering problem of the CDGSKE --- p.130 / Chapter 4. --- The BT for the λ-mCDGSKE --- p.135 / Chapter 5. --- The BT for the CDGSKE --- p.136 / Chapter 6. --- References --- p.139 / Summary --- p.140

by Zheng Yu-kun. / Thesis (M.Ph.)--Chinese University of Hong Kong, 1987. / Includes bibliographies.

Submitted by Nathaly Infantini Spano (natyspano@gmail.com) on 2018-04-11T22:15:07Z No. of bitstreams: 1 Nathaly Spano.pdf: 542621 bytes, checksum: 9cce243dcd5f335d0ff41628f513a126 (MD5) / Approved for entry into archive by Hellen Sayuri Sato null (hellen@ift.unesp.br) on 2018-04-12T17:27:02Z (GMT) No. of bitstreams: 1 spano_ni_dr_ift.pdf: 542621 bytes, checksum: 9cce243dcd5f335d0ff41628f513a126 (MD5) / Made available in DSpace on 2018-04-12T17:27:02Z (GMT). No. of bitstreams: 1 spano_ni_dr_ift.pdf: 542621 bytes, checksum: 9cce243dcd5f335d0ff41628f513a126 (MD5) Previous issue date: 2018-02-27 / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / A integrabilidade da hierarquia de Korteweg de-Vries modificada supersimétrica com N=1 (smKdV) na presença de defeitos é investigada através da construção de sua transformação de Bäcklund supersimétrica. A construção de tal transformação é realizada usando essencialmente dois métodos: a abordagem da matriz de defeito e empregando o operador de recursão. Primeiramente, empregamos a matriz de defeitos associada à hierarquia, que é a mesma para o modelo sinh-Gordon supersimétrico (sshG). O método é geral e válido para todos os fluxos da hierarquia e como exemplo derivamos explicitamente as equações de Bäcklund para os primeiros fluxos, são eles t_1, t_3 e t_5. Em segundo lugar, o operador de recursão relacionando tempos consecutivos é derivado e mostrados que ele relaciona também as transformações de Bäcklund. Finalmente, esta transformação de Bäcklund supersimétrica é empregada para introduzir defeitos do tipo I para a hierarquia supersimétrica mKdV. Outros aspectos de integrabilidade são considerados, através da construção das quantidades conservadas modificadas, derivadas da matriz de defeito. / The integrability of the N=1 supersymmetric modified Korteweg de-Vries (smKdV) hierarchy in the presence of defects is investigated through the construction of its super Bäcklund transformation. The construction of such transformation is performed by essentially using two methods: the Bäcklund-defect matrix approach and the by employing the recursion operator. Firstly, we employ the defect matrix associated to the hierarchy which turns out to be the same for the supersymmetric sinh-Gordon (sshG) model. The method is general for all flows and as an example we derive explicitly the Bäcklund equations in components for the first few flows of the hierarchy, namely t_1, t_3 and t_5. Secondly, the recursion operator relating consecutive time flows is derived and shown to relate their Bäcklund transformations. Finally, this super Bäcklund transformation is employed to introduce type I defects for the supersymmetric mKdV hierarchy. Further integrability aspects by considering modified conserved quantities are derived from the defect matrix. / CNPq: 141204/2014-1

In this thesis, we study a Bäcklund transformation for minimal surfaces - surfaces with vanishing mean curvature - transforming a given minimal surface into a possible infinity of new ones. The transformation, also carrying with it mappings between solutions to the elliptic Liouville equation, is first derived by using geometrical concepts, and then by using algebraic methods alone - the latter we have not been able to find elsewhere. We end by exploiting the transformation in an example, transforming the catenoid into a family of new minimal surfaces.

published_or_final_version / Physics / Master / Master of Philosophy

Este Trabalho apresenta novas soluções exatas para as equações de Navier – Stokes transientes tridimensionais para escoamentos viscosos incompressíveis. Estas soluções são obtidas por meio de Split e Transformações Auto-Bäcklund. O procedimento de Split desacopla as equações de Navier – Stokes em dois sistemas de equações diferenciais parciais, um linear e outro não-linear, ambos não-homogêneos. O sistema linear, que contém somente termos viscosos e derivadas temporais, é resolvido via Transformações Auto-Bäcklund induzidas por relações de comutação, fornecendo o campo de velocidades. Os componentes do vetor velocidade são então substituídos no sistema não-linear a fim de obter o correspondente campo de pressões. A resolução do sistema não-linear para a pressão pode ser obtida tanto numericamente (via integração direta) quanto analiticamente, empregando a equação de Helmholtz. O objetivo do presente trabalho é encontrar expressões analíticas para o campo de velocidades e obter resultados numéricos para o campo de pressão associado. O caráter híbrido das soluções proporciona uma redução significativa do tempo de processamento requerido para a simulação de escoamentos viscosos, o qual praticamente se reduz ao tempo demandado para a tarefa de pós-processamento. Com esse objetivo em mente, foi desenvolvida uma formulação tridimensional escalar para a função corrente, a fim de reduzir o tempo requerido na tarefa mais dispendiosa de pós-processamento, a saber, o traçado das linhas de corrente em torno de corpos submersos de formato arbitrário. Neste estágio de desenvolvimento, esta formulação é empregada para produzir mapas de linhas de corrente para escoamentos viscosos em torno de uma esfera para números de Reynolds elevados. / This work presents new exact solutions to the unsteady three dimensional Navier-Stokes equations for incompressible viscous flows. These solutions are obtained by means of split and auto-Bäcklund transformations. The splitting procedure decouples the Navier-Stokes equations into a linear and a nonlinear inhomogeneous system of partial differential equations. The linear system, which contains only viscous terms and time derivatives, is solved via auto-Bäcklund transformations induced by commutation relations, furnishing the velocity field. The components of the velocity vector are then replaced into the nonlinear system to obtain the corresponding pressure field. The solution of the nonlinear system for the pressure variable can be carried out either numerically (by direct integration) or analytically, using the Helmholtz equation . The aim of the proposed work is to find analytical expressions for the velocity field and to obtain numerical results to the associated pressure field. The hybrid character of the solutions provides a significant reduction on the time processing required to simulate viscous flows, which virtually reduces to the time demanded to execute post-processing tasks. Taking this fact in mind, a three dimensional scalar formulation for the streamfunction was developed in order to simplify the most time-consuming post-processing task required, e.g., plotting the streamlines around arbitrary shaped bodies. At this stage of development, this formulation is employed to produce streamline maps for viscous flows around a sphere for high Reynolds numbers.

Este Trabalho apresenta novas soluções exatas para as equações de Navier – Stokes transientes tridimensionais para escoamentos viscosos incompressíveis. Estas soluções são obtidas por meio de Split e Transformações Auto-Bäcklund. O procedimento de Split desacopla as equações de Navier – Stokes em dois sistemas de equações diferenciais parciais, um linear e outro não-linear, ambos não-homogêneos. O sistema linear, que contém somente termos viscosos e derivadas temporais, é resolvido via Transformações Auto-Bäcklund induzidas por relações de comutação, fornecendo o campo de velocidades. Os componentes do vetor velocidade são então substituídos no sistema não-linear a fim de obter o correspondente campo de pressões. A resolução do sistema não-linear para a pressão pode ser obtida tanto numericamente (via integração direta) quanto analiticamente, empregando a equação de Helmholtz. O objetivo do presente trabalho é encontrar expressões analíticas para o campo de velocidades e obter resultados numéricos para o campo de pressão associado. O caráter híbrido das soluções proporciona uma redução significativa do tempo de processamento requerido para a simulação de escoamentos viscosos, o qual praticamente se reduz ao tempo demandado para a tarefa de pós-processamento. Com esse objetivo em mente, foi desenvolvida uma formulação tridimensional escalar para a função corrente, a fim de reduzir o tempo requerido na tarefa mais dispendiosa de pós-processamento, a saber, o traçado das linhas de corrente em torno de corpos submersos de formato arbitrário. Neste estágio de desenvolvimento, esta formulação é empregada para produzir mapas de linhas de corrente para escoamentos viscosos em torno de uma esfera para números de Reynolds elevados. / This work presents new exact solutions to the unsteady three dimensional Navier-Stokes equations for incompressible viscous flows. These solutions are obtained by means of split and auto-Bäcklund transformations. The splitting procedure decouples the Navier-Stokes equations into a linear and a nonlinear inhomogeneous system of partial differential equations. The linear system, which contains only viscous terms and time derivatives, is solved via auto-Bäcklund transformations induced by commutation relations, furnishing the velocity field. The components of the velocity vector are then replaced into the nonlinear system to obtain the corresponding pressure field. The solution of the nonlinear system for the pressure variable can be carried out either numerically (by direct integration) or analytically, using the Helmholtz equation . The aim of the proposed work is to find analytical expressions for the velocity field and to obtain numerical results to the associated pressure field. The hybrid character of the solutions provides a significant reduction on the time processing required to simulate viscous flows, which virtually reduces to the time demanded to execute post-processing tasks. Taking this fact in mind, a three dimensional scalar formulation for the streamfunction was developed in order to simplify the most time-consuming post-processing task required, e.g., plotting the streamlines around arbitrary shaped bodies. At this stage of development, this formulation is employed to produce streamline maps for viscous flows around a sphere for high Reynolds numbers.

Este Trabalho apresenta novas soluções exatas para as equações de Navier – Stokes transientes tridimensionais para escoamentos viscosos incompressíveis. Estas soluções são obtidas por meio de Split e Transformações Auto-Bäcklund. O procedimento de Split desacopla as equações de Navier – Stokes em dois sistemas de equações diferenciais parciais, um linear e outro não-linear, ambos não-homogêneos. O sistema linear, que contém somente termos viscosos e derivadas temporais, é resolvido via Transformações Auto-Bäcklund induzidas por relações de comutação, fornecendo o campo de velocidades. Os componentes do vetor velocidade são então substituídos no sistema não-linear a fim de obter o correspondente campo de pressões. A resolução do sistema não-linear para a pressão pode ser obtida tanto numericamente (via integração direta) quanto analiticamente, empregando a equação de Helmholtz. O objetivo do presente trabalho é encontrar expressões analíticas para o campo de velocidades e obter resultados numéricos para o campo de pressão associado. O caráter híbrido das soluções proporciona uma redução significativa do tempo de processamento requerido para a simulação de escoamentos viscosos, o qual praticamente se reduz ao tempo demandado para a tarefa de pós-processamento. Com esse objetivo em mente, foi desenvolvida uma formulação tridimensional escalar para a função corrente, a fim de reduzir o tempo requerido na tarefa mais dispendiosa de pós-processamento, a saber, o traçado das linhas de corrente em torno de corpos submersos de formato arbitrário. Neste estágio de desenvolvimento, esta formulação é empregada para produzir mapas de linhas de corrente para escoamentos viscosos em torno de uma esfera para números de Reynolds elevados. / This work presents new exact solutions to the unsteady three dimensional Navier-Stokes equations for incompressible viscous flows. These solutions are obtained by means of split and auto-Bäcklund transformations. The splitting procedure decouples the Navier-Stokes equations into a linear and a nonlinear inhomogeneous system of partial differential equations. The linear system, which contains only viscous terms and time derivatives, is solved via auto-Bäcklund transformations induced by commutation relations, furnishing the velocity field. The components of the velocity vector are then replaced into the nonlinear system to obtain the corresponding pressure field. The solution of the nonlinear system for the pressure variable can be carried out either numerically (by direct integration) or analytically, using the Helmholtz equation . The aim of the proposed work is to find analytical expressions for the velocity field and to obtain numerical results to the associated pressure field. The hybrid character of the solutions provides a significant reduction on the time processing required to simulate viscous flows, which virtually reduces to the time demanded to execute post-processing tasks. Taking this fact in mind, a three dimensional scalar formulation for the streamfunction was developed in order to simplify the most time-consuming post-processing task required, e.g., plotting the streamlines around arbitrary shaped bodies. At this stage of development, this formulation is employed to produce streamline maps for viscous flows around a sphere for high Reynolds numbers.

Conservation laws play an important role in science. The aim of this thesis is to provide an overview and develop new methods for constructing conservation laws using Lie group theory. The derivation of conservation laws for invariant variational problems is based on Noether’s theorem. It is shown that the use of Lie-Bäcklund transformation groups allows one to reduce the number of basic conserved quantities for differential equations obtained by Noether’s theorem and construct a basis of conservation laws. Several examples on constructing a basis for some well-known equations are provided. Moreover, this approach allows one to obtain new conservation laws even for equations without Lagrangians. A formal Lagrangian can be introduced and used for computing nonlocal conservation laws. For self-adjoint or quasi-self-adjoint equations nonlocal conservation laws can be transformed into local conservation laws. One of the fields of applications of this approach is electromagnetic theory, namely, nonlocal conservation laws are obtained for the generalized Maxwell-Dirac equations. The theory is also applied to the nonlinear magma equation and its nonlocal conservation laws are computed.