Orientadores: Rui Loja Fernandes, Luiz Antonio Barrera San Martin / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-12T16:18:57Z (GMT). No. of bitstreams: 1 Struchiner_Ivan_D.pdf: 1576350 bytes, checksum: 7c87189c22a89931d1a38ac563188723 (MD5) Previous issue date: 2009 / Resumo: O objetivo desta tese é mostrar como utilizar algebróides de Lie e grupóides de Lie para compreender aspectos das teorias de invariantes, simetrias e espaços de moduli de estruturas geométricas de tipo finito. De uma forma geral, podemos descrever tais estruturas como sendo objetos, definidos em uma variedade, que podem ser caracterizados por correferenciais (possivelmente em outra variedade). Exemplos incluem G-estruturas de tipo finito e geometrias de Cartan. Para uma classe de estruturas geométricas de tipo finito cujo espaço de moduli (dos germes) de seus elementos tem dimensão finita, construímos um algebróide de Lie A X, chamado de algebróide de Lie classificante, que satisfaz as seguintes propriedades: 1. Para cada ponto na base X corresponde um germe de uma estrutura geométrica pertencente à classe. 2. Dois destes germes são isomorfos se e somente se eles correspondem ao mesmo ponto de X. 3. A álgebra de Lie de isotropia de A num ponto x é a álgebra de Lie das simetrias infinitesimais da estrutura geométrica correspondente. 4. Se dois germes de estruturas geométricas pertencem à mesma estrutura geométrica global numa variedade conexa, então eles correspondem a pontos na mesma órbita de A em X. Além do mais, quando o algebróide de Lie classificante é integrável, o seu grupóide de Lie pode ser utilizado para construir modelos explícitos das geometrias na classe sendo descrita. Estes modelos são universais, ou seja, qualquer outra estrutura geométrica da classe é localmente isomorfa a um destes modelos, e globalmente equivalentes, a menos de recobrimento, a um subconjunto aberto de um desses modelos. No caso em que a estrutura geométrica é uma G-estrutura de tipo finito, damos uma descrição detalhada dessa correspondência. Uma das conseqüências da nossa construção é que o algebróide de Lie classificante pode ser usado para obter invariantes das estruturas geométricas correspondentes. Para ilustrar, apresentamos dois exemplos de invariantes que são induzidos pela cohomologia do algebróide de Lie. Para demonstrar os resultados mencionados acima, definimos as noções de forma de Maurer-Cartan em grupóides de Lie e de equação de Maurer-Cartan para um formas diferenciais com valores num algebróide de Lie. A seguir, provamos que a forma de Maurer-Cartan em um grupóide de Lie satisfaz uma propriedade universal análoga à propriedade satisfeita pela forma de Maurer-Cartan em um grupo de Lie. Para concluir esta tese, descrevemos diversos exemplos relacionados as conexões sem torção em G-estruturas. Nossa classe principal de exemplos são as conexões simpléticas especiais para as quais incluímos uma discussão detalhada. / Abstract: The purpose of this thesis is to show how to use Lie algebroids and Lie groupoids to get a better understanding of problems concerning symmetries, invariants and moduli spaces of geometric structures of finite type. In general terms, these structures are objects defined on manifolds which can be characterized by a coframe (on a possibly different manifold). Examples include G-structures of finite type and Cartan geometries. For a given class of such structures whose moduli space (of germs) of elements is finite dimensional, we are able to construct a Lie algebroid A ! X, called the classifying Lie algebroid, which has the following properties: 1. To each point on the base X there corresponds a germ of a geometric structure which belongs to the class. 2. Two such germs are isomorphic if and only if they correspond to the same point in X. 3. The isotropy Lie algebra of A at a point x is the symmetry Lie algebra of the corresponding geometric structure. 4. If two germs of the geometric structure belong to the same connected manifold, then they correspond to points on the same orbit of A in X. Moreover, when the classifying Lie algebroid is integrable, its Lie groupoid can be used to construct explicit models of the geometries in the class being described. These models turn out to be universal in the sense that every other geometric structure in the class is locally isomorphic to one of these models, and globally equivalent up to covering to an open set of one of these models. We describe this throughly when the geometric structure in consideration is a finite type G-structure. One of the consequences of our construction is that the classifying Lie algebroid can be used to obtain invariants of the corresponding geometric structures. We present two examples of invariants that are induced by the cohomology of the Lie algebroid. The method that we use to prove the statements above is to define the notion of a Maurer-Cartan form on a Lie groupoid, as well as a Maurer-Cartan equation for Lie algebroid valued differential one forms. We then prove a universal property for the Maurer-Cartan form of a Lie groupoid. We believe that these results are of independent interest. To conclude this thesis, we give a description of several examples related to torsionfree connections on G-structures. Our main class of examples are the special symplectic connections for which we include a detailed discussion. / Doutorado / Geometria Diferencial / Doutor em Matemática

Este trabalho aborda a formulação geométrica das teorias clássicas de calibre, ou Yang-Mills, considerando-as como uma importante classe de modelos que deve ser incluída em qualquer tentativa de estabelecer um formalismo matemático geral para a teoria clássica dos campos. Tal formulação deve vir em (pelo menos) duas variantes: a versão hamiltoniana, que passou por uma fase de desenvolvimento rápido durante os últimos 10-15 anos, levando ao que hoje é conhecido como o ``formalismo multissimplético\'\', e a mais tradicional versão lagrangiana utilizada nesta tese. O motivo principal justificando tal investigação é que teorias de calibre constituem os mais importantes exemplos de sistemas dinâmicos que são altamente relevantes na Física e onde a equivalência entre a versão lagrangiana e a versão hamiltoniana, que no caso de sistemas não-singulares é estabelecida pela transformação de Legendre, deixa de ser óbvia, pois teorias de calibre são sistemas degenerados do ponto de vista lagrangiano e são sistemas vinculados do ponto de vista hamiltoniano. Esta propriedade característica das teorias de calibre é uma consequência direta do seu alto grau de simetria, isto é, da sua invariância de calibre. No entanto, numa formulação plenamente geométrica da teoria clássica dos campos, capaz de incorporar situações topologicamente não-triviais, a invariância sob transformações de calibre locais (transformações de calibre de segunda espécie) e, surpreendentemente, até mesmo a invariância sob as transformações de simetria globais correspondentes (transformações de calibre de primeira espécie) não podem ser adequadamente descritas em termos de grupos de Lie e suas ações em variedades, mas requerem a introdução e o uso sistemático de um novo conceito, a saber, fibrados de grupos de Lie e suas ações em fibrados (sobre a mesma variedade base). A meta principal da presente tese é tomar os primeiros passos no desenvolvimento de ferramentas matemáticas adequadas para lidar com este novo conceito de simetria e, como uma primeira aplicação, dar uma definição clara e simples do procedimento de ``acoplamento mínimo\'\' e uma demonstração simples do teorema de Utiyama, segundo o qual lagrangianas para potenciais de calibre (conexões) de primeira ordem (i.e., que dependem apenas dos próprios potenciais de calibre e de suas derivadas parciais até primeira ordem) que são invariantes sob transformações de calibre são necessariamente funções dos campos de calibre (i.e., do tensor de curvatura) invariantes sob as transformações de simetria globais correspondentes. / This thesis deals with the geometric formulation of classical gauge theories, or Yang-Mills theories, regarded as an important class of models that must be included in any attempt to establish a general mathematical framework for classical field theory. Such a formulation must come in (at least) two variants: the hamiltonian version which has gone through a phase of rapid development during the last 10-15 years, leading to what is now known as the ``multisymplectic formalism\'\', and the more traditional lagrangian version studied in this thesis. The main motivation justifying this kind of investigation is that gauge theories constitute the most important examples of dynamical systems that are highly relevant in physics and where the equivalence between the lagrangian and the hamiltonian version, which for non-singular systems is established through the Legendre trans% formation, is far from obvious, since gauge theories are degenerate systems from the lagrangian point of view and are constrained systems from the hamiltonian point of view. This characteristic property of gauge theories is a direct consequence of their high degree of symmetry, that is, of gauge invariance. However, in a fully geometric formulation of classical field theory, capable of incorporating topologically non-trivial situations, invariance under local gauge transformations (gauge transformations of the second kind) and, surprisingly, even invariance under the corresponding global symmetry transformations (gauge transformations of the first kind) cannot be described adequately in terms of Lie groups and their actions on manifolds but requires the introduction and systematic use of a new concept, namely Lie group bundles and their actions on fiber bundles (over the same base manifold). The main goal of the present thesis is to take the first steps in developing adequate mathematical tools for handling this new concept of symmetry and, as a first application, give a simple clear-cut definition for the prescription of ``minimal coupling\'\' and a simple proof of Utiyama´s theorem, according to which lagrangians for gauge potentials (connections) that are gauge invariant and of first order, i.e., dependent only on the gauge potentials themselves and on their partial derivatives up to first order, are necessarily functions of the gauge field strengths (i.e., the curvature tensor) invariant under the corresponding global symmetry transformations.

Submitted by Renata Lopes (renatasil82@gmail.com) on 2017-06-27T19:48:01Z No. of bitstreams: 1 guilhermemarquesribeiro.pdf: 1025565 bytes, checksum: 681a6e31eb787e74456258f00aa65e53 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2017-08-07T21:07:11Z (GMT) No. of bitstreams: 1 guilhermemarquesribeiro.pdf: 1025565 bytes, checksum: 681a6e31eb787e74456258f00aa65e53 (MD5) / Made available in DSpace on 2017-08-07T21:07:11Z (GMT). No. of bitstreams: 1 guilhermemarquesribeiro.pdf: 1025565 bytes, checksum: 681a6e31eb787e74456258f00aa65e53 (MD5) Previous issue date: 2015-07-31 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Nessa dissertação, apresentamos dois formalismos consistentes para tratar a dinâmica de sistemas vinculados: o procedimento de Dirac[15], baseado num algoritmo que substitui os parênteses de Poisson por outra estrutura semelhante, e o método simplético[17], fundamentado na deformação da estrutura simplética do espaço de fase. Aplicamos esses formalismos tanto em exemplos simples quanto em problemas concretos de física teórica, como o modelo de Proca e o campo eletromagnético. Estudamos também as simetrias apresentadas por sistemas vinculados de primeira classe. Apresentamos uma prova da conjectura de Dirac[3] e mostramos que um contra-exemplo apresentado na literatura[2] é consistente com a conjectura . / In this dissertation, we have presented two consistent formalisms to treat the dynamics of constrained systems: the Dirac procedure[15], based on an algorithm that replaces the Poisson brackets by a similar structure, and the symplectic method[17], based on the deformation of the symplectic structure of the phase space. We have applied this formalisms to both simple examples and concrete problems from theoretical physics, such as the Proca model and the electromagnetic field. We also studied the symmetries generated by first class constrained systems. We have presented a prove of Dirac's conjecture[3] and showed that a counter-example found in the literature[2] is consistent with the conjecture

Les propriétés optiques linéaires et non-linéaires de nanoparticules métalliques de tailles non-négligeables comparées à celles des longueurs d’onde excitatrices sont étudiées dans cette thèse. Les informations issues de la symétrie sont mises en avant afin de décrire des nanoparticules appartenant à des groupes ponctuels. Pour cela, un formalisme totalement irréductible est mis en place afin de prendre en compte l’extension spatiale des objets étudiés. Dans ce formalisme, le tenseur de réponse non-linéaire possède un nombre fini de valeurs significatives reliant les composantes multipolaires des champs incidents et sortants. Ce formalisme est alors appliqué analytiquement à l’étude de la réponse non- linéaire du second ordre de nano-étoiles d’or en interprétant des mesures de SHG résolue en polarisation. Finalement, des expériences de spectroscopies multidimensionnelles sont utilisées dans le but de connecter les propriétés spatiales et les propriétés spectrales de ces objets. L’introduction de modes propres définis par la symétrie des objets permet encore une fois de donner un sens physique aux comportements électroniques mis en jeu / Using metallic nanoparticles with a threefold symmetry thorough the study, the impact of the symmetry on the nonlinear properties is investigated. Interpretations of polarization-resolved SHG experiments indicate the importance of multipolar resonances, in particular quadrupole and octupole, to explain the strong values of the nonlinear susceptibilities in such systems. A fully irreducible formalism is then developed to treat extended objects like nanoparticles. In this formalism, the nonlinear response tensor is a discrete set of values easily constrained by symmetries instead of a field. This formalism permits to describe simply linear and nonlinear optical response from nanoparticles. Finally, time-domain experiments are conducted with the aim to connect spatial and spectral properties. These experiments allow to interpret the spectra in terms of eigenmodes

There are several methods avaliable to find areas of interest, but most fail at detecting such areas in cluttered scenes. In this paper two methods will be presented and tested in a qualitative perspective. The first is the darg operator, which is used to detect three dimensional convex or concave objects by calculating the derivative of the argument of the gradient in one direction of four rotated versions. The four versions are thereafter added together in their original orientation. A multi scale version is recommended to avoid the problem that the standard deviation of the Gaussians, combined with the derivatives, controls the scale of the object, which is detected. Another feature detected in this paper is rotational symmetries with the help of approximative polynomial expansion. This approach is used in order to minimalize the number and sizes of the filters used for a correlation of a representation of the orientation and filters matching the rotational symmetries of order 0, 1 and 2. With this method a particular type of rotational symmetry can be extracted by using both the order and the orientation of the result. To improve the method’s selectivity a normalized inhibition is applied on the result, which causes a much weaker result in the two other resulting pixel values when one is high. Both methods are not enough by themselves to give a definite answer to if the image consists of an area of interest or not, since several other things have these types of features. They can on the other hand give an indication where in the image the feature is found.

In this thesis, several new aspects of asymptotic symmetries have been exploited.Firstly, we have shown that the asymptotic symmetries can be enhanced tosymplectic symmetries in three dimensional asymptotically Anti-de Sitter (AdS) space-time with Dirichletboundary conditions. Such enhancement providesa natural connection between the asymptotic symmetries in the far region i.e. closeto the boundary) and the near-horizon region, which leads to a consistenttreatment for both cases. The second investigation in three dimensional space-time is to study theEinstein-Maxwell theory including asymptotic symmetries, solutionspace and surface charges with asymptotically flat boundary conditionsat null infinity. This model allows one to illustrate several aspectsof the four dimensional case in a simplified setting. Afterwards, we givea parallel analysis of Einstein-Maxwell theory in the asymptotically AdScase.Another new aspect consists in demonstrating a deep connection between certainasymptotic symmetry and soft theorem. Recently, a remarkable equivalence wasfound between the Ward identity of certain residual (large) U(1) gauge transformations and the leadingpiece of the soft photon theorem. It is well known that the softphoton theorem includes also a sub-leading piece. We have proven thatthe large U(1) gauge transformation responsible for the leading soft factorcan also explain the sub-leading one.In the last part of the thesis, wewill investigate the asymptotic symmetries near the inner boundary. Asa null hypersurface, the black hole horizon can be considered as an innerboundary. The near horizon symmetries create “soft” degrees of freedom. Wehave generalised such argument to isolated horizon and have shown that those “soft” degreesof freedom of an isolated horizon are equivalent to its electric multipolemoments. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

The standard model of particle physics is the most precisely verified scientific theory in the history of mankind. However, extended theories are already in place, ready to supersede the standard model should it fail to describe any new physics that may be observed in the next generation of high energy particle accelerators. One such minimal extension is the Two Higgs Doublet Model (2HDM). However, the appearance of additional symmetries to those of the gauge symmetries in the 2HDM can have consequences for the cosmological viability of the model, with the possibility for non-trivial topological defects forming during spontaneous symmetry breaking phase transitions.In this research we perform a systematic study of six accidental Higgs Family and CP symmetries that can occur in the 2HDM potential, by introducing and utilizing our Majorana scalar-field formalism. General sufficient conditions for convexity and stability of the scalar potential are derived and analytical solutions for two non-zero neutral vacuum expectation values of the Higgs doublets for each of the six symmetries are presented, in terms of the parameters of the theory. We identify the topological defects associated with the spontaneous symmetry breaking of each symmetry by means of a homotopy-group analysis. We find the existence of domain walls from the breaking of Z2, CP1 and CP2 discrete symmetries, vortices in models with broken U(1)PQ and CP3 symmetries and a global monopole in the SO(3)HF-broken model. We study the associated topological defect solutions as functions of the potential parameters via gradient flow methods. We also consider the cosmological implications of the topological defects and are able to derive bounds on physical observables of the theory in order to avoid contradictions with the theoretical limits on topological defects. The application of our Majorana scalar-field formalism in studying more general scalar potentials that are not constrained by the U(1)Y hypercharge symmetry is discussed. In particular, the formalism may be used to properly identify seven previously hidden symmetries that may be manifest in a U(1)Y invariant scalar potential for particular choices of the model parameters.

No / The purpose of this paper is to highlight the importance of constraints in the theory of relativity and, in particular, what philosophical work they do for Einstein's views on the laws of physics. Einstein presents a view of local ``structure laws'' which he characterizes as the most appropriate form of physical laws. Einstein was committed to a view of science, which presents a synthesis between rational and empirical elements as its hallmark. If scientific constructs are free inventions of the human mind, as Einstein, held, the question arises how such rational constructs, including the symbolic formulation of the laws of physics, can represent physical reality. Representation in turn raises the question of realism. Einstein uses a number of constraints in the theory of relativity to show that by imposing constraints on the rational elements a certain ``fit'' between theory and reality can be achieved. Fit is to be understood as satisfaction of constraint. His emphasis on reference frames in the STR and more general coordinate systems in the GTR, as well as his emphasis on the symmetries of the theory of relativity suggests that Einstein's realism is akin to a certain form of structural realism. His version of structural realism follows from the theory of relativity and is independent of any current philosophical debates about structural realism.

Στην παρούσα διατριβή παρουσιάζεται η μελέτη μιας οικογένειας εξισώσεων διαφορών (ή διακριτών εξισώσεων) χρησιμοποιώντας μεθόδους συμμετριών. Τέτοιες μέθοδοι είναι καλά θεμελιωμένες για την μελέτη και κατασκευή λύσεων διαφορικών εξισώσεων. Στόχος είναι η χρήση συμμετριών για τη σύνδεση διαφορικών και διακριτών εξισώσεων, καθώς και η κατασκευή λύσεων των τελευταίων από συμμετρικές λύσεις των πρώτων. Συγκεκριμένα, μελετάμε διακριτές εξισώσεις που είναι αφινικά γραμμικές, έχουν τις συμμετρίες του τετραγώνου και εμπλέκουν τέσσερεις τιμές μιας άγνωστης συνάρτησης δύο ακέραιων μεταβλητών, οι οποίες σχηματιζούν ένα στοιχειώδες τετράπλευρο στο επίπεδο των ανεξάρτητων μεταβλητών. Η διεξοδική μελέτη αυτής της οικογένειας οδηγεί στην κατασκευή ενός νόμου διατήρησης καθώς και σε συνθήκες γραμμικοποιήσης. Μέλη αυτής της οικογένειας είναι και οι ολοκληρώσιμες εξισώσεις της ταξινόμησης των Adler, Bobenko, Suris (ABS). Η ολοκληρωσιμότητα των εξισώσεων ABS προκύπτει από την πολυδιάστατη συμβατότητά τους. Αυτό σημαίνει ότι μπορούν να επεκταθούν κατάλληλα σε εξισώσεις πολλών ανεξάρτητων μεταβλητών. Η ιδιότητα αυτή μας επιτρέπει να κατασκευάσουμε άμεσα έναν αυτομεταχηματισμό Bäcklund και ένα ζευγάρι Lax χρησιμοποιώντας τις ίδιες τις εξισώσεις, στοιχεία που αποτελούν άλλη μια ένδειξη της ολοκληρωσιμότητάς τους. Η εξάρτηση των εξισώσεων ABS από δύο συνεχείς παραμέτρους μας επιτρέπει να μελετήσουμε επιπλέον και τις επεκταμένες συμμετρίες τους, δηλαδή τις συμμετρίες που δρουν και στις παραμέτρους. Αυτές οι συμμετρίες αποτελούν το βασικό εργαλείο για τη σύνδεσή τους με ολοκληρώσιμα συστήματα διαφορικών εξισώσεων. Την ολοκληρωσιμότητα αυτών των συμβατών διαφορικών συστημάτων την αποδεικνύουμε κατασκευάζοντας έναν αυτομετασχηματισμό Bäcklund και ένα ζευγάρι Lax. Η σύνδεση αυτή μας επιτρέπει να κατασκευάσουμε λύσεις των διακριτών εξισώσεων από λύσεις του συμβατού συστήματος διαφορικών εξισώσεων, οι οποίες συνδέονται με λύσεις των συνεχών εξισώσεων Painlevé. Από την άλλη, παρουσιάζεται η σύνδεση αυτών των συστημάτων διαφορικών εξισώσεων με τις γεννήτριες εξισώσεις. Οι τελευταίες παρουσιάστηκαν αρχικά από τους Nijhoff, Hone, Joshi χρησιμοποιώντας άλλη προσέγγιση. Ωστόσο, η προσέγγιση μέσω συμμετρικών αναγωγών που παρουσιάζουμε εδώ είναι πιο άμεση και οδηγεί στα ίδια συμπεράσματα. Συνοψίζοντας, η παρούσα διατριβή παρουσιάζει μια καινοτομική χρήση των συμμετριών των διακριτών εξισώσεων για την κατασκευή λύσεων, αλλά και την σύνδεσή τους με συστήματα διαφορικών εξισώσεων. / In the present dissertation, we present the study of a family of discrete equations using symmetry-based techniques. Such methods are well established for the study of differential equations. We use the symmetries of discrete equations to establish new connections between discrete and differential equations, as well as to construct new solutions of the former in terms of similarity solutions of the latter. Specifically, we study discrete equations which are affine linear, possess the symmetries of the square and involve four values of an unknown function of two independent discrete variables forming a quadrilateral. The extensive study of this class leads to a conservation law, as well as to linearization conditions. Members of this family are the integrable equations of the Adler, Bobenko, Suris (ABS) classification. The integrability of the ABS equations follows from their multidimensional consistency. The latter implies that, the equation may be extended in a multidimensional lattice. This property allows us to derive directly an auto– Bäcklund transformation and a Lax pair, using the function defining these equations. These are another evidence of the integrability of the ABS equations. The dependence of these equations on two continuous parameters permits us to study their extended symmetries, i.e. symmetries acting on the parameters as well. These symmetries are our main tool in connecting the ABS equations to integrable systems of differential equations. The integrability of the latter is proved by the construction of an auto–Bäcklund transformation and a Lax pair. This connection provides us the means to construct solutions of the discrete equations from solutions of the compatible differential system, which are related to solutions of the continuous Painlevé equations. On the other hand, we present how these systems lead naturally to generating differential equations, which were presented by Nijhoff, Hone and Joshi starting from another point of view. However, our construction through symmetry reductions is more straightforward. Thus, in the present thesis is presented a novel usage of the symmetries of discrete equations in the construction of solutions and the connection between discrete and differential equations.

Ce mémoire est une partie d’un programme de recherche qui étudie la superintégrabilité des systèmes avec spin. Plus particulièrement, nous nous intéressons à un hamiltonien avec interaction spin-orbite en trois dimensions admettant une intégrale du mouvement qui est un polynôme matriciel d’ordre deux dans l’impulsion. Puisque nous considérons un hamiltonien invariant sous rotation et sous parité, nous classifions les intégrales du mouvement selon des multiplets irréductibles de O(3). Nous calculons le commutateur entre l’hamiltonien et un opérateur général d’ordre deux dans l’impulsion scalaire, pseudoscalaire, vecteur et pseudovecteur. Nous donnons la classification complète des systèmes admettant des intégrales du mouvement scalaire et vectorielle. Nous trouvons une condition nécessaire à remplir pour le potentiel sous forme d’une équation différentielle pour les cas pseudo-scalaire et pseudo-vectoriel. Nous utilisons la réduction par symétrie pour obtenir des solutions particulières de ces équations. / This thesis is part of a research program studying superintegrable systems with spin. In particular, we consider a Hamiltonian with a spin-orbital interaction in three dimensions admitting an integral of motion that is a matrix polynomial second order in the momenta. Since we are considering a Hamiltonian which is invariant under rotation and parity, we classify the integrals of motion into irreducible O(3) multiplets. We obtain the commutator of the Hamiltonian with the scalar, pseudoscalar, vector and axial vector operators. We provide a complete classification for the scalar and vector cases. We find the necessary condition for superintegrability on the potential as a differential equation. We use symmetry reduction methods to obtain particular solutions of this equation.