Spatiotemporal properties of classical coupled nonlinear oscillators are investigated in this thesis. Chapter 1 gives an introduction to nonlinear lattices and to the concept of breathers, that are spatially localized and temporally periodic excitation in nonlinear lattices. The concept of anti-continuous limit that provides the basic methodology in probing spatiotemporal breather properties is discussed. In Chapter 2, the general approach for finding exact breather solutions from the anti-continuous limit is examined, and the rotating wave approximation(RWA) is applied to probe the spatial structure of static breathers. Numerical evidence reveals that the RWA relates the spatial structure of stable multi-breathers to a single breather of the same frequency. Chapter 3 presents linear stability analysis of static breathers and gives a systematic way to construct mobile breathers. Formation and collision properties of this moving breathers are also studied. Chapter 4 discusses dynamics of kinks and anti-kinks in hydrogen-bonded chains in the context of two-component soliton model. From molecular dynamics simulations with finite temperature, it is observed that, in a real system (eg. ice), a pair of kink and anti-kink can evolve into a moving-breather-like excitation. Chapter 5 is devoted to the understand of the effects of disorder in the Holstein model. The summary is given in Chapter 6.

We address the excitation of quantum breathers in small nonlinear networks of two and three degrees of freedom, in order to study their properties. The invariance under permutation of two sites of these networks substitutes the translation invariance that is present in nonlinear lattices, where (classical) discrete breathers are time periodic space localized solutions of the underlying classical equations of motion. We do a systematic analysis of the spectrum and eigenstates of such small systems, characterizing quantum breather states by their tunnelling rate (energy splitting), site correlations, fluctuations of the number of quanta, and entanglement. We observe how these properties are reflected in the time evolution of initially localized excitations. Quantum breathers manifest as pairs of nearly degenerate eigenstates that show strong site correlation of quanta, and are characterized by a strong excitation of quanta on one site of the network which perform slow coherent tunnelling motion from one site to another. They enhance the fluctuations of quanta, and are the least entangled states among the group of eigenstates in the same range of the energy spectrum. We use our analysis methods to consider the excitation of quantum breathers in a cell of two coupled Josephson junctions, and study their properties as compared with those in the previous cases. We describe how quantum breathers could be experimentally observed by employing the already developed techniques for quantum information processing with Josephson junctions.

In the mid-1920s, the great Albert Einstein proposed that at extremely low temperatures, a gas of bosonic particles will enter a new phase where a large fraction of them occupy the same quantum state. This state would bring many of the peculiar features of quantum mechanics, previously reserved for small samples consisting only of a few atoms or molecules, up to a macroscopic scale. This is what we today call a Bose-Einstein condensate. It would take physicists almost 70 years to realize Einstein's idea, but in 1995 this was finally achieved. The research on Bose-Einstein condensates has since taken many directions, one of the most exciting being to study their behavior when they are placed in optical lattices generated by laser beams. This has already produced a number of fascinating results, but it has also proven to be an ideal test-ground for predictions from certain nonlinear lattice models. Because on the other hand, nonlinear science, the study of generic nonlinear phenomena, has in the last half century grown out to a research field in its own right, influencing almost all areas of science and physics. Nonlinear localization is one of these phenomena, where localized structures, such as solitons and discrete breathers, can appear even in translationally invariant systems. Another one is the (in)famous chaos, where deterministic systems can be so sensitive to perturbations that they in practice become completely unpredictable. Related to this is the study of different types of instabilities; what their behavior are and how they arise. In this thesis we compare classical and quantum mechanical nonlinear lattice models which can be applied to BECs in optical lattices, and also examine how classical nonlinear concepts, such as localization, chaos and instabilities, can be transfered to the quantum world.

We address the excitation of quantum breathers in small nonlinear networks of two and three degrees of freedom, in order to study their properties. The invariance under permutation of two sites of these networks substitutes the translation invariance that is present in nonlinear lattices, where (classical) discrete breathers are time periodic space localized solutions of the underlying classical equations of motion. We do a systematic analysis of the spectrum and eigenstates of such small systems, characterizing quantum breather states by their tunnelling rate (energy splitting), site correlations, fluctuations of the number of quanta, and entanglement. We observe how these properties are reflected in the time evolution of initially localized excitations. Quantum breathers manifest as pairs of nearly degenerate eigenstates that show strong site correlation of quanta, and are characterized by a strong excitation of quanta on one site of the network which perform slow coherent tunnelling motion from one site to another. They enhance the fluctuations of quanta, and are the least entangled states among the group of eigenstates in the same range of the energy spectrum. We use our analysis methods to consider the excitation of quantum breathers in a cell of two coupled Josephson junctions, and study their properties as compared with those in the previous cases. We describe how quantum breathers could be experimentally observed by employing the already developed techniques for quantum information processing with Josephson junctions.

[EN] Nature is nonlinear. The linear description of physical phenomena is useful for explain observations with the simplest mathematical models, but they are only accurate for a limited range of input values. In the case of intense acoustics waves, linear models obviate a wide range of physical phenomena that are necessary for accurately describe such high-amplitude waves, indispensable for explain other exotic acoustic waves and mandatory for developing new applied techniques based on nonlinear processes. In this Thesis we study the interactions between nonlinearity and other basic wave phenomena such as non-classical attenuation, anisotropic dispersion and periodicity, and diffraction in specific configurations. First, we present intense strain waves in a chain of cations coupled by realistic interatomic potentials. Here, the nonlinear ionic interactions and lattice dispersion lead to the formation of supersonic kinks. These intrinsically-nonlinear localized dislocations travel long distances without changing its properties and explain the formation of dark traces in mica crystals. Then, we analyze nonlinear wave processes in a system composed of multilayered acoustic media. The rich nonlinear dynamics of this system is characterized by its strong dispersion. Here, harmonic generation processes and the relation with its band structure are presented, showing that the nonlinear processes can be enhanced, strongly minimized or simply modified by tuning the layer parameters. In this way, we show how the dynamics of intense monochromatic waves and acoustic solitons can be controlled by artificial layered materials. In a second part, we include diffraction and analyze four types of singular beams. First, we study nonlinear beams in two dimensional sonic crystals. In this system, the inclusion of anisotropic dispersion is tuned for obtain simultaneous self-collimation for fundamental and second harmonic beams. The conditions for optimal second harmonic generation are presented. Secondly, we present limited diffraction beam generation using equispaced axisymmetric diffraction gratings. The obtained beams are truncated version of zero-th order Bessel beams. Third, the grating spacing can be modified to achieve focusing, where the generated nonlinear beams presents high gain, around 30 dB, with a focal width which is between the diffraction limit and the sub-wavelength regime, but with its characteristic high amplitude side lobes strongly reduced. Finally, we observe that waves diffracted by spiral-shaped gratings generate high-order Bessel beams, conforming nonlinear acoustic vortex. The conditions to obtain arbitrary-order Bessel beams by these passive elements are presented. Finally, the interplay of nonlinearity and attenuation in biological media is studied in the context of medical ultrasound. First, a numerical method is developed. The method solves the constitutive relations for nonlinear acoustics and the frequency power law attenuation of biological media is modeled as a sum of relaxation processes. A new technique for reducing numerical dispersion based on artificial relaxation is included. Second, this method is used to study the harmonic balance as a function of the power law, showing the role of weak dispersion and its impact on the efficiency of the harmonic generation in soft-tissues. Finally, the study concerns the nonlinear behavior of acoustic radiation forces in frequency power law attenuation media. We present how the interplay between nonlinearity and the specific frequency power law of biological media can modify the value for acoustic radiation forces. The relation of the nonlinear acoustic radiation force with thermal effects are also discussed. The broad range of nonlinear processes analyzed in this Thesis contributes to understanding the behavior of intense acoustic waves traveling trough complex media, while its implications for enhancing existent applied acoustics techniques are presented. / [ES] La Naturaleza es no lineal. La descripción lineal de los fenómenos físicos es de gran utilidad para explicar nuestras observaciones con modelos matemáticos simples, pero éstos sólo son precisos en un limitado rango de validez. En el caso de onda acústica de alta intensidad, los modelos lineales obvian un amplio rango de fenómenos físicos que son necesarios para describir con precisión las ondas de gran amplitud, pero además son necesarios para explicar otros procesos más exóticos e indispensables para desarrollar nuevas aplicaciones basadas en propagación no lineal. En esta Tesis, estudiamos las interacciones entre no linealidad y otros procesos complejos como atenuación no-clásica, dispersión anisotrópica y periodicidad, y difracción en configuraciones específicas. En primer lugar, presentamos ondas de deformación en una cadena de cationes acoplados por potenciales realísticas. Aquí, las interacciones no lineales entre iones, producen la conformación de kinks supersónicos. Estas dislocaciones localizadas intrínsecamente no lineales viajan por la red largas distancias sin variar sus propiedades, y pueden explicar la formación de trazas en minerales como la mica. Aumentando la escala del problema, estudiamos los procesos acústicos no lineales en medios multicapa. La rica dinámica de estos medios está caracterizada por la fuerte dispersión debido a la periodicidad del sistema. Aquí, estudiamos los procesos de generación de harmónicos, mostrando como modificando la estructura podemos potenciar, minimizar, o simplemente modificar artificialmente la transferencia de energía entre las componentes espectrales, y de esta manera controlar la dinámica de las ondas y solitones en el interior de la estructura. En la segunda parte, incluimos difracción y analizamos cuatro tipos de haces singulares. En primer lugar, analizamos haces ultrasónicos no lineales en cristales de sonido bidimensionales. En este sistema, las propiedades de anisotropía del medio son ajustadas para obtener la auto-colimación simultánea del primer y segundo harmónico. Así, se obtiene la propagación no difractiva para las dos componentes. En segundo lugar, presentamos haces de difracción limitada empleando rejillas de difracción axisimétricas. Por último, demostramos la generación de haces de Bessel de orden superior mediante estructuras en espiral. En la última parte, estudiamos la competición entre no linealidad y la atenuación y dispersión observable en medios biológicos en el contexto de las aplicaciones de biomédicas de los ultrasonidos. En primer lugar desarrollamos un nuevo método computacional para la dependencia frecuencial en forma de ley de potencia de la absorción característica de los tejidos. Este método en dominio temporal es usado posteriormente para revisar los procesos básicos no lineales prestando especial interés en el paper de la dispersión del tejido. Por último, la resolución de las ecuaciones constitutivas nos permite abordar la descripción no lineal de la fuerza de radiación acústica producida en tejidos biológicos, y las implicaciones existentes con la deposición de energía y transferencia de momento para ondas ultrasónicas de alta intensidad. El amplio abanico de procesos no lineales analizados en esta tesis contribuye a una mejor comprensión de la dinámica de las ondas acústicas de alta intensidad en medios complejos, donde las implicaciones existentes en cuanto a la mejora de sus aplicaciones prácticas son puestas de manifiesto. / [CA] La Naturalesa és no lineal. La descripció lineal dels fenòmens físics és de gran utilitat per a explicar les nostres observacions amb models matemàtics simples, però aquests sol són precisos en un limitat rang de validesa. En el cas d'ona acústica d'alta intensitat, els models lineals obvien un ampli rang de fenòmens físics que són necessaris per a descriure amb precisió les ones de gran amplitud, però a més són necessaris per a explicar altres processos més exòtics i indispensables per a desenvolupar noves aplicacions basades en propagació no lineal. En aquesta Tesi, estudiem les interaccions entre no-linealitat i altres processos complexos com atenuació no-clàssica, dispersió anisotròpica i periodicitat, i difracció en configuracions específiques. En primer lloc, presentem ones de deformació en una cadena de cations acoblats per potencials realistes. Ací, les interaccions no lineals entre ions, produeixen la conformació de kinks supersònics. Aquestes dislocacions localitzades intrínsecament no lineals viatgen per la xarxa llargues distàncies sense variar les seues propietats, i poden explicar la formació de traces en minerals com la mica. Augmentant l'escala del problema, estudiem els processos acústics no lineals en mitjans multicapa. La rica dinàmica d'aquests mitjans es caracteritza per la forta dispersió a causa de la periodicitat del sistema. Ací, estudiem els processos de generació d'harmònics, mostrant com modificant l'estructura podem potenciar, minimitzar, o simplement modificar artificialment la transferència d'energia entre les components espectrals, i d'aquesta manera controlar la dinàmica de les ones i solitons a l'interior de l'estructura. En la segona part, incloem difracció i analitzem quatre tipus de feixos singulars. En primer lloc, analitzem feixos ultrasònics no lineals en cristalls de so bidimensionals. En aquest sistema, les propietats d'anisotropia del medi són ajustades per a obtenir l'acte-col·limació simultània del primer i segon harmònic. Així, s'obté la propagació no difractiva per a les dues components. En segon lloc, presentem feixos de difracció limitada emprant reixetes de difracció axisimètriques. Per últim, vam demostrar la generació de feixos de Bessel d'ordre superior mitjançant estructures en espiral. En l'última part, estudiem la competició entre no linealitat i l'atenuació i dispersió observable en medis biològics en el context de les aplicacions biomèdiques dels ultrasons. En primer lloc desenvolupem un nou mètode computacional per a la dependència freqüencial en forma de llei de potència de l'absorció característica dels teixits biològics. Aquest mètode en domini temporal és usat posteriorment per a revisar els processos bàsics no lineals prestant especial interés en el paper de la dispersió del teixit. Per últim, la resolució de les equacions constitutives ens permet abordar la descripció no lineal de la força de radiació acústica produïda en teixits biològics, i les implicacions existents amb la deposició d'energia i transferència de moment per a ones ultrasòniques d'alta intensitat. L'ampli ventall de processos no lineals analitzats en aquesta tesi contribueix a una millor comprensió de la dinàmica de les ones acústiques d'alta intensitat en medis complexos, on les implicacions existents quant a la millora de les seues aplicacions practiques són posades de manifest. / Jiménez González, N. (2015). Nonlinear Acoustic Waves in Complex Media [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/53237 / Premios Extraordinarios de tesis doctorales