The recent empirical literature has shown that the dynamic generating mechanism of macroeconomic variables can be asymmetric. Inspiring from these empirical results, this thesis uses a class of nonlinear models called smooth transition autoregressive models to investigate possible asymmetric dynamics in inflation and nominal exchange rate series of Turkey. Estimation results imply that variables under consideration contain strong nonlinearities and these can be modeled by STAR models.

In this thesis, we work on the upper bound for nonlinearity of t-resilient Boolean functions given by Sarkar and Maitra, which is based on divisibility properties of spectral weights of resilient functions and study construction methods that achieve the upper bound. One of the construction methods, introduced by Maity and Johansson, starts with a bent function and complements some values of its truth table corresponding to a previously chosen set of inputs, S, which satisfies three criteria. In this thesis, we show that a fourth criterion is needed for t-resiliency of the resulting function, and prove that three criteria of Maity and Johansson do not guarantee resiliency. We also work on other constructions, one by Sarkar and Maitra, which uses a Maiorana-McFarland like technique to satisfy the upper bound and the other by Tarannikov, which satisfies the nonlinearity bound using a technique with low computational complexity. However, these methods have tendency to maximize the order of resiliency for a given number of variables, therefore one cannot construct functions for all possible resiliency values given the number of variables, using this method. We further go into details and compute the auto-correlation functions of the constructed Boolean functions to find the absolute indicator and sum-of-squared-errors for each of them. We also provide a comparison of Boolean functions constructed by other techniques given in the literature, together with the ones studied in this thesis.

This dissertation consists of three main parts. The first part is to discuss aileron reversal problem for a typical section with linear aerodynamic and structural analysis. The result gives some insight and ideas for this aeroelastic problem. Although the aileron in its post-reversal state will work the opposite of its design, this type of phenomenon as a design root should not be ruled out on these grounds alone, as current active flight-control systems can compensate for this. Moreover, one can get considerably more (negative) lift for positive flap angle in this unusual regime than positive lift for positive flap angle in the more conventional setting. This may have important implications for development of highly maneuverable aircraft. The second part is to involve the nonlinear aerodynamic and structural analyses into the aileron reversal problem. Two models, a uniform cantilevered lifting surface and a rolling aircraft with rectangular wings, are investigated here. Both models have trailing-edge control surfaces attached to the main wings. A configuration that reverses at a relatively low dynamic pressure and flies with the enhanced controls at a higher level of effectiveness is demonstrated. To evaluate how reliable for the data from XFOIL, the data for the wing-aileron system from advanced CFD codes and experiment are used to compare with that from XFOIL. To enhance rolling maneuverability for an aircraft, the third part is to search for the optimal configuration during the post-reversal regime from a design point of view. Aspect ratio, hinge location, airfoil dimension, inner structure of wing section, composite skin, aeroelastic tailoring, and airfoil selection are investigated for cantilevered wing and rolling aircraft models, respectively. Based on these parametric structural designs as well as the aerodynamic characteristics of different airfoils, recommendations are given to expand AAW flight program.

Boolean functions are an important area of study for cryptography. These functions, consisting merely of one's and zero's, are the heart of numerous cryptographic systems and their ability to provide secure communication. Boolean functions have application in a variety of such systems, including block ciphers, stream ciphers and hash functions. The continued study of Boolean functions for cryptography is therefore fundamental to the provision of secure communication in the future. This thesis presents an investigation into the analysis of Boolean functions and in particular, analysis of affine transformations with respect to both the design and application of Boolean functions for cryptography. Past research has often been limited by the difficulties arising from the magnitude of the search space. The research presented in this thesis will be shown to provide an important step towards overcoming such restrictions and hence forms the basis for a new analysis methodology. The new perspective allows a reduced view of the Boolean space in which all Boolean functions are grouped into connected equivalence classes so that only one function from each class need be established. This approach is a significant development in Boolean function research with many applications, including class distinguishing, class structures, self mapping analysis and finite field based s-box analysis. The thesis will begin with a brief overview of Boolean function theory; including an introduction to the main theme of the research, namely the affine transformation. This will be followed by the presentation of a fundamental new theorem describing the connectivity that exists between equivalence classes. The theorem of connectivity will form the foundation for the remainder of the research presented in this thesis. A discussion of efficient algorithms for the manipulation of Boolean functions will then be presented. The ability of Boolean function research to achieve new levels of analysis and understanding is centered on the availability of computer based programs that can perform various manipulations. The development and optimisation of efficient algorithms specifically for execution on a computer will be shown to have a considerable advantage compared to those constructed using a more traditional approach to algorithm optimisation. The theorem of connectivety will be shown to be fundamental in the provision many avenues of new analysis and application. These applications include the first non-exhaustive test for determining equivalent Boolean functions, a visual representation of the connected equivalence class structure to aid in the understanding of the Boolean space and a self mapping constant that enables enumeration of the functions in each equivalence class. A detailed survey of the classes with six inputs is also presented, providing valuable insight into their range and structure. This theme is then continued in the application Boolean function construction. Two important new methodologies are presented; the first to yield bent functions and the second to yield the best currently known balanced functions of eight inputs with respect to nonlinearity. The implementation of these constructions is extremely efficient. The first construction yields bent functions of a variety of algebraic order and inputs sizes. The second construction provides better results than previously proposed heuristic techniques. Each construction is then analysed with respect to its ability to produce functions from a variety of equivalence classes. Finally, in a further application of affine equivalence analysis, the impact to both s-box design and construction will be considered. The effect of linear redundancy in finite field based s-boxes will be examined and in particular it will be shown that the AES s-box possesses complete linear redundancy. The effect of such analysis will be discussed and an alternative construction to s-box design that ensures removal of all linear redundancy will be presented in addition to the best known example of such an s-box.

Fundamental to the electronic security of information and communication systems, is the correct use and application of appropriate ciphers. The strength of these ciphers, particularly in their ability to resist cryptanalytic attacks, directly in uences the overall strength of the entire system. The strength of the underlying cipher is reliant upon a robust structure and the carefully designed interaction between components in its architecture. Most importantly, however, cipher strength is critically dependent on the strength of the individual components of which it is comprised. Boolean functions and substitution boxes (s-boxes) are among the most common and essential components of ciphers. This is because they are able to provide a cipher with strengthening properties to resist known and potential cryptanalytic attacks. Thus, it is not surprising that significant research effort has been made in trying to develop ways of obtaining boolean functions and substitution boxes with optimal achievable measures of desirable cryptographic properties. Three of the main cryptographic properties required by strong boolean functions and s-boxes are nonlinearity, correlation immunity and propagation criteria, with different cryptographic applications requiring different acceptable measures of these and other properties. As combinations of cryptographic properties exhibited by functions can be conicting, finding cryptographically strong functions often means that a trade-off needs to be made when optimizing property values. Throughout this thesis, the term "optimization" specifically refers to seeking to obtain the best achievable combination of target property values which may be exhibited by boolean functions and s-boxes, regardless of whether the relevant properties are conflicting or complementary. This thesis focusses on a particular class of techniques for obtaining strong functions for cryptographic applications, referred to as heuristic methods or, simply, heuristics. Three new heuristic methods, each aimed at generating boolean functions optimizing one or more of the main cryptographic properties mentioned above, in addition to other desirable properties, are presented. The first of the new heuristic methods developed for this thesis focusses on generating boolean functions which are balanced and exhibit very high nonlinearities. Highly nonlinear balanced functions are critical to many cryptographic applications, as they provide good resistance to linear cryptanalytic attacks. This first method is based on the recursive modification of a starting bent function and is shown to be highly successful and efficient at generating numerous such functions, which also exhibit low autocorrelation values, in a very short computational time. The generation of balanced, correlation immune boolean functions that also exhibit the confl icting property of high nonlinearity is the focus of the second new heuristic method developed for this thesis. By concatenating selected pairs of lower-dimensional boolean functions together in the Walsh Hadamard transform domain, direct optimization for both resilience and nonlinearity was able to take place at each level towards and for the final function. This second method was able to generate examples of boolean functions with almost all of the best known optimal combinations of target property values. Experiments have shown the success of this method in consistently generating highly nonlinear resilient boolean functions, for a range of orders of resilience, with such functions possessing optimal algebraic degree. A third new heuristic method, which searches for balanced boolean functions which satisfy a non-zero degree of propagation criteria and exhibit high nonlinearity, is presented. Intelligent bit manipulations in the truth table of starting functions, based on fundamental relationships between boolean function transforms and measures, provide the design rationale for this method. Two new function generation schemes have been proposed for this method, to efficiently satisfy the requirements placed on the starting functions utilized in the computational process. An optional process attempts to increase the algebraic degree of the resulting functions, without sacrificing the optimalities that are achievable. The validity of this method is demonstrated through the success of various experimental trials. Switching the focus from single output boolean functions to multiple output boolean functions (s-boxes), the effectiveness of existing heuristic techniques (namely Genetic Algorithm, Hill Climbing Method and combined Genetic Algorithm/Hill Climbing) in primarily being applied to improve the nonlinearity of s-boxes of various dimensions, is investigated. The prior success of these heuristic techniques for improving the nonlinearity of boolean functions has been previously demonstrated, as has the success of hill climbing in isolation when applied to bijective s-boxes. An extension to the bijective s-box optimization work is presented in this thesis. In this new research, a Genetic Algorithm, Hill Climbing Method and the two in combination are applied to the nonlinearity and autocorrelation optimization of regular NxM s-boxes (N > M) to investigate the effectiveness and efficiency of each of these heuristics. A new breeding scheme, utilized in the Genetic Algorithm and combined Genetic Algorithm/Hill Climbing trials, is also presented. The success of experimental results compared to random regular s-box generation is demonstrated. New research in applying the Hill Climbing Method to construct NxM sboxes (N > M) required to meet specific property criteria is presented. The consideration of the characteristics desired by the constructed s-boxes largely dictated the generation process. A discussion on the generation process of the component functions is included. Part of the results produced by experimental trials were incorporated into a commonly used family of stream ciphers, thus further supporting the use of heuristic techniques as a useful means of obtaining strong functions suitable for incorporation into practical ciphers. An analysis of the cryptographic properties of the s-box used in the MARS block cipher, the method of generation and the computational time taken to obtain this s-box, led to the new research reported in this thesis on the generation of MARS-like s-boxes. It is shown that the application of the Hill Climbing Method, with suitable requirements placed on the component boolean functions, was able to generate multiple MARS-like s-boxes which satisfied the MARS sbox requirements and provided additional properties. This new work represented an alternative approach to the generation of s-boxes satisfying the MARS sbox property requirements but which are cryptographically superior and can be obtained in a fraction of the time than that which was taken to produce the MARS s-box. An example MARS-like s-box is presented in this thesis. The overall value of heuristic methods in generating strong boolean functions and substitution boxes is clearly demonstrated in this thesis. This thesis has made several significant contributions to the field, both in the development of new, specialized heuristic methods capable of generating strong boolean functions, and in the analysis and optimization of substitution boxes, the latter achieved through applying existing heuristic techniques.

[Truncated abstract] Runoff generation behaviour and flooding in a lake dominated catchment are nonlinear, threshold-driven processes that result from the interactions between climate and various catchment characteristics. A complicating feature of the rainfall to runoff transformation, which may have implications for the flood frequency, is that the various surface and subsurface flow pathways are dynamic, heterogeneous and highly nonlinear, consisting of distinct thresholds. To understand the impact of threshold nonlinearities on the rainfall-runoff transformation in such catchments, a systematic examination was carried out to investigate runoff generation behaviour of the catchment itself, the overflow behaviour of a lake in combination with the catchment draining into it, as well as the lake organisation within a lake chain network. Three storage based thresholds were considered: the catchment field capacity storage governing catchment subsurface stormflow, total storage capacity governing catchment surface runoff, and lake storage capacity governing lake-overflow. ... Through these investigations, this thesis has provided valuable insights into the process controls of lake-overflow events and the associated flood frequency behaviour in lake dominated catchments. In particular, the relative roles of climate, soil depth, the soil?s drainage capacity, as well as the relative geometry of the lake vis a vis the contributing catchment, in the determination of the dynamic characteristics of lake-overflow events and associated flood frequency behaviour have been highlighted. In addition, the importance of lake organization, as expressed in terms of the average ratio of catchment area to lake area and the spatial variability of this ratio from upstream to downstream, and their impact upon connectivity and flood frequency have also been explored. The outcomes of this study highlight the importance of thresholds governing flood frequency, and provide insights into the complex interactions between rainfall variability and the various threshold nonlinearities in the rainfall-runoff process, which are shown to have a significant impact on the resulting flood frequency curves. The improved understanding of these process controls will be useful in assisting the 1 management of the catchment-lake system in the study region, and in regions elsewhere. In particular, the outcome of this study can provide guidance towards the adoption of various management strategies for lake chain systems by illustrating the effects of potential flow interruption and retardation as ways to assist in flood prevention and mitigation, whether it is aimed at decreasing the frequency of occurrence of lake overflows, or merely decreasing the flow magnitude for a given return period.

In this PhD thesis we consider the effect of nonlinearity and correlated disorder in lowdimensional electronic systems. We study the dynamics of one electron wave packet in a chain with a nonadiabatic electron-phonon interaction. The electron-phonon coupling is taken into account in the time-dependent Schr¨odinger equation by a delayed cubic nonlinearity. In the limit of an adiabatic coupling, the self-trapping phenomenon occurs when the nonlinearity parameter exceeds a critical value of the order of the bandwidth. We show that a weaker nonlinearity is required to produce self-trapping in the regime of short delay times. However, this trend is reversed for slow nonlinear responses, resulting in a reentrant phase diagram. In slowly responding media, self-trapping only takes place for very strong nonlinearities. In additional we consider the two-dimensional Anderson model with correlated disorder. We bring our attention to an Anderson model with diagonal disorder diluted by an underlying periodicity. The model consists of two interpenetrating sublattices, one composed of random potentials (Anderson lattice) and the other composed of non-random constant potentials. By using numerical calculations we compute the localization properties and provide a accurate estimates of the level spacing distribution. Within our numerical precision we demonstrate that the states close to band center obey a Wigner-Like distribution. For strong disorder a intermediate Poisson-Wigner-Like level spacing distribution is obtained. / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Nesta tese de doutorado queremos considerar efeitos de não linearidade e desordem correlacionada em sistemas eletrônicos em baixa dimensionalidade. Nós estudamos a dinâmica de um pacote de onda eletrônico em uma cadeia com uma interação elétron-fônon não adiabática. O acoplamento elétron-fônon considerado na equação de Schrödinger dependente do tempo por uma não-linearidade cúbica retardada. No limite de um acoplamento adiabático, o fenômeno de auto-aprisionamento ocorre quando o parâmetro de não linearidade excede um valor crítico da ordem da largura da banda. Nó mostramos que uma não-linearidade fraca é necessária para produzir auto-aprisionamento em regime de tempos curtos de retardo. No entanto, esta tendência é invertida para resposta não linear lenta, resultando em um diagrama de fase reentrante. Em meios que respondem lentamente, o auto-aprisionamento surge apenas para não linearidades muito fortes. Em adicional, nós trazemos o modelo de Anderson bidimensional com desordem correlacionada. Voltamos nossa atenção para um modelo com desordem diagonal diluída por uma periodicidade subjacente. O modelo consiste de duas subredes interpenetrantes, uma composta de potenciais aleatórios (rede de Anderson) e a outra composta de potenciais não aleatórios constantes. Usando cálculos numéricos nós calculamos as propriedades de localização e fornecemos uma estimativa da distribuição dos espaçamentos de níveis. Dentro da nossa precisão numérica nós demonstramos que os estados próximos ao centro da banda obedecem a uma distribuição do tipo Wigner. Para desordem fortes uma distribuição dos espaçamentos de níveis intermediária do tipo Poisson-Wigner é obtida.

Dans cette thèse, nous étudions des systèmes nano-électro-mécaniques (NEMS) en conditions cryogéniques (de 30 mK à 30 K) sensibles à des conditions de fluctuations ou de désordre. Les phénomènes fondamentaux étudiés sont omniprésents dans la physique des NEMS, et pour certains vont même au-delà avec des analogies vers d’autres disciplines de la physique, comme les transitions de phase ou la RMN.Dans la première partie de cette thèse, nous nous intéressons ainsi au bruit d'amplitude du NEMS, fournissant un exemple de mouvement Brownien dans un potentiel de confinement. Du fait de la non-linéarité géométrique intrinsèque au système, l'anharmonicité du potentiel transduit le mouvement Brownien de chaque mode mécanique en fluctuations des fréquences propres de résonance. Ainsi, nous observons expérimentalement un phénomène de diffusion spectrale, se traduisant par un élargissement et un décalage de la raie de résonance non-triviaux rendant compte de la compétition entre la diffusion de la phase de la réponse du mode due à la transduction, et les mécanismes de relaxation du mode fluctuant. Une approche par intégrale de chemin de la diffusion capture l'effet analytiquement. Un tel mécanisme altère la résonance d'un mode mécanique sans influer sur les échanges d'énergie avec le bain thermodynamique du mode. En outre, l'introduction d'une forte excitation sinusoïdale agit en retour sur les fluctuations hors équilibre via la non-linéarité, ralentissant la dynamique du système et comprimant ses fluctuations pour certains points critiques de l'espace des paramètres, près du ou dans le régime de la réponse bistable permise par la non-linéarité. Enfin, des expériences-modèles ont été réalisées afin de comprendre en détail la décohérence mécanique classique à l’aide d’un bruit en fréquence extrinsèque, réalisé à l’aide d’une grille couplée au NEMS.La deuxième partie de cette thèse explore plus en détail certains mécanismes microscopiques de relaxation d'énergie ou du bruit en fréquence interne d’un mode mécanique, encore partiellement incompris pour les NEMS. Nous considérons d’abord le cas d’une contribution extérieure, mais universelle, qui a pour origine le transfert d’impulsion entre le NEMS et le gaz présent dans la cellule expérimentale, ici l’hélium 4. Dans la limite des faibles densités, la théorie cinétique décrit la dissipation dans le gaz ballistique. De façon inattendue, nous observons aux plus basses pressions atteignables une déviation à la théorie. Nous montrons pour plusieurs températures et plusieurs échantillons que cette déviation s’échelonne avec le rapport entre le libre parcours moyen des atomes dans le gaz et la hauteur du NEMS vis-à-vis du fond de l’échantillon. Ce résultat est justifié par un modèle phénoménologique prenant en compte la réflexion diffusive des atomes du gaz sur le mur du fond, qui présente à petite échelle une structure désordonnée. Cette réflexion résulte en une déviation à la Maxwellienne près du fond, et donc en l’établissement d’un gradient de densité du gaz sur une longueur de l’ordre du libre parcours moyen, qui renormalise le taux de relaxation d’énergie mécanique. Ainsi, le NEMS agit comme une sonde non-invasive d’un milieu hors équilibre du fait de ses très petites dimensions transverses. Enfin, nous mesurons la dissipation intrinsèque du NEMS jusqu’à 30 milliKelvin. Nous mettons en évidence le rôle des excitations de basse énergie couplées à la déformation du NEMS dans la relaxation d’énergie mécanique. Ces excitations, permises par la structure désordonnée des matériaux constitutifs du NEMS, sont modélisées comme des atomes se déplaçant par effet tunnel entre deux positions équivalentes du réseau atomique (TLS). Nous obtenons également le bruit en fréquence intrinsèque en développant une nouvelle technique de mesure utilisant la non-linéarité du NEMS. L’étude poussée nous permet de lier phénoménologiquement les deux phénomènes. / In this thesis we address cryogenic nano-electro-mechanical systems (NEMS) from 30 mK to 30 K sensitive to conditions involving fluctuations or disorder. The fundamental aspects studied are ubiquitous in NEMS physics, and for some of them go beyond, with possible analogies with phase transitions or NMR.In the first part of this work we focus on the NEMS position noise, which is a good example of Brownian motion within a confinement potential. Owing to the system’s intrinsic geometric nonlinearity, the potential anharmonicity translates each mode’s Brownian motion into fluctuations of the structure’s resonance eigenfrequencies. As a result we observe experimentally a spectral diffusion phenomenon that manifests through a linewidth broadening and a frequency shift of the resonance line: they account non-trivially for the competition between the probed mode’s response’s phase diffusion due to the transduction mechanism and the fluctuating modes relaxation mechanisms. A path integral approach to diffusion encompasses analytically the effect. Such a mechanism alters a mechanical mode’s resonance without changing energy transfers to the mode’s thermal bath. Furthermore, adding a strong sinusoidal excitation acts back on the out-of-equilibrium fluctuations through the nonlinearity: the system dynamics is slowed down, with its fluctuations squeezed, in peculiar points of the parameters space, near or within the non-linearity induced bistable regime. Finally, model experiments are realized so as to understand classical mechanical decoherence, through the use of an extrinsic frequency noise, artificially crafted thanks to a gate electrode coupled to the NEMS.In a second part, some microscopic mechanisms leading to mechanical damping and internal frequency noise of a mechanical mode are investigated, as they are still elusive to date for NEMS. We first consider the case of an external but universal source of damping, which originates from the momentum transfer between the NEMS and the gas flowing in the experimental cell, here Helium 4. In the rarefied limit, dissipation in a ballistic gas is well described by kinetic theory. Yet, unexpectedly, we observe at our lowest pressures a discrepancy between our measurements and theory. We show for several temperatures and samples that this deviation scales with the ratio between the gas atoms mean free path and the gap between the NEMS and the sample’s bottom trench. This result is modelled phenomenologically as arising from diffusive scattering of gas atoms at the bottom’s wall, which at small lengthscales has a disordered landscape. Diffusive scattering results in a deviation to the Maxwellian distribution, leading to a gas density gradient in the vicinity of the wall, established over a distance comparable with the mean free path, and which renormalizes the mechanical energy relaxation rate. Therefore, the NEMS acts as a non-invasive probe in a nonequilibrium medium due to its small cross-section. Finally, we investigate the NEMS intrinsic dissipation down to 30 milliKelvin. We highlight the role of low-energy excitations coupled to the NEMS deformation in damping mechanisms. These excitations, allowed by the disordered structure of the NEMS constitutive materials, are modelled as atoms tunneling between two equivalent positions of the atomic lattice (also referred to as TLS). Using a new technique which relies on the NEMS non-linearity, we measure the intrinsic frequency noise, and we show that it can be linked phenomenologically to the damping due to the TLS.

Submitted by Luciana Ferreira (lucgeral@gmail.com) on 2018-07-17T13:07:11Z No. of bitstreams: 2 Tese - Luciano Calaça Alves - 2018.pdf: 4699371 bytes, checksum: 706846314ebb74648be890b03e18d4cc (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2018-07-17T13:55:46Z (GMT) No. of bitstreams: 2 Tese - Luciano Calaça Alves - 2018.pdf: 4699371 bytes, checksum: 706846314ebb74648be890b03e18d4cc (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2018-07-17T13:55:46Z (GMT). No. of bitstreams: 2 Tese - Luciano Calaça Alves - 2018.pdf: 4699371 bytes, checksum: 706846314ebb74648be890b03e18d4cc (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2018-06-07 / Fundação de Amparo à Pesquisa do Estado de Goiás - FAPEG / This work presents the study of solitary wave solutions, known as solitons, in non-linear and non- homogeneous media using non-linear Schrödinger equations. Three cases are studied: first considering a logarithmic nonlinear term; second with saturation effect and finally including effects of high orders (Raman scattering). Solutions are modulated by three different types of potential. First, linear in the spatial and oscillatory coordinate in the temporal coordinate. The second, quadratic in the spatial and oscillatory in the temporal coordinates. Finally, it is also modulated using a mixed potential, which is the junction of the two potentials presented above. After including inomogeneities in linear and nonlinear coefficients, the similarity transformation technique is used to convert the non-linear, non-autonomous equation into an autonomous one that will be solved analytically. This field of study has potential applications in crystals, optical fibers and in Bose- Einstein condensates, also serving to understand the fundamentals related to this state of matter. The stability of the solutions are checked by numerical simulations. / Este trabalho apresenta o estudo de soluções de ondas solitárias, conhecidas como sólitons, em meios não lineares e não homogêneos por meio de equações não lineares de Schrödinger. São estudados três casos: primeiro considerando um termo não linear do tipo logarítmico; segundo com efeito de saturação e por último incluindo efeitos de altas ordens (espalhamento Raman). As soluções são moduladas por três tipos diferentes de potencial. O primeiro, linear na coordenada espacial e oscilatório na coordenada temporal. O segundo, quadrático na coordenada espacial e oscilatório na temporal. Por fim, modula-se também utilizando um potencial misto, que é a junção dos dois potenciais apresentados anteriormente. Depois de incluir heterogeneidades nos coeficientes lineares e não lineares, é utilizada a técnica da transformação de similaridade para converter a equação não linear, não autônoma em uma autônoma que será resolvida analiticamente. Esse campo de estudo tem potenciais aplicações em cristais, fibras ópticas e em condensados de Bose-Einstein, servindo também para o entendimento dos fundamentos relacionados a esse estado da matéria. A estabilidade das soluções são checadas por meio de simulações numéricas.

Nos últimos anos grande número de modelos computacionais tem sido propostos com o intuito de descrever o movimento de solutos no perfil do solo, apesar disso, o que se observa é que existe grande dificuldade em se modelar esses fenômenos, para que o modelo possa predizer o processo de deslocamento e retenção dos solutos na natureza. Para tanto, o objetivo deste trabalho foi utilizar um modelo estatístico para descrever o transporte dos solutos no perfil do solo. Dessa forma, foi realizado um experimento em laboratório e observado os níveis de potássio e nitrato ao longo do perfil dos solos Latossolo Vermelho Amarelo e Nitossolo Vermelho. Para inferir sobre essas variáveis foram consideradas duas abordagens. Para a primeira abordagem foi utilizado um modelo de regressão não linear para cada uma das variáveis, cujos parâmetros do modelo apresentam uma interpretação prática, na área de solos. Para esse modelo foi realizado um esboço sobre a não linearidade do mesmo para verificar as propriedades assintóticas dos estimadores dos parâmetros. Para o método de estimação foi considerado, o método de mínimos quadrados e o método de bootstrap. Além disso, foi realizada uma análise de diagnóstico para verificar a adequação do modelo, bem como identificar pontos discrepantes. Por outro lado, para outra abordagem, foi utilizado um modelo multiresposta para analisar o comportamento das variáveis nitrato e potássio ao longo do perfil dos solos, conjuntamente. Para esse modelo foi utilizado o método da máxima verossimilhança para encontrar as estimativas dos parâmetros do modelo. Em ambas as situações, observou-se a adequação dos modelos para descrever o comportamento dos solutos nos solos, sendo uma alternativa para os pesquisadores que trabalham com estudo de solos. O modelo logístico com quatro parâmetros se destacou por apresentar melhores propriedades, como medidas de não linearidade e boa qualidade de ajuste. / In the last years, several computational models have been proposed to describe the movement of solutes in the soil profile, but what is observed is that there is great difficulty in model these phenomena, so that model can predict the displacement process and retention of solutes in nature. Thus, the aim of this study was to use a statistical model to describe the transport of solutes in the soil profile. Therefore, an experiment was conducted in the laboratory and observed levels of potassium and nitrate along the depth of soil Oxisol (Haplustox) and Hapludox,. To make inferences about these variables were considered two approaches. For the first approach was utilized a non-linear regression model for each variable and the model parameters have a practical interpretation on soil. For this model we performed a sketch on the nonlinearity of the model to check the asymptotic properties of parameter estimators. To estimate the parameters were considered the least squares method and the bootstrap method. In addition, we performed a diagnostic analysis to verify the adequacy of the model and identify outliers. In the second approach considered was using a multi-response model to analyze the behavior of the variables nitrate and potassium throughout the soil profile together. For this model we used the maximum likelihood method to estimate the model parameters. In both cases, we observed the suitability of the models to describe the behavior of solutes in soils, being an alternative for researchers working on the study of soils. The logistic model with four parameters stood out with better properties, such as non-linearity and good fit.