Nonlinear Supersymmetry as a Hidden Symmetry

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Integrability, rational solitons and symmetries for nonlinear systems in Biology and Materials Physics

[ES] Los sistemas no lineales constituyen un tema de investigación de creciente interés en las últimas décadas dada su versatilidad en la descripción de fenómenos físicos en diversos campos de estudio. Generalmente, dichos fenómenos vienen modelizados por ecuaciones diferenciales no lineales, cuya estructura matemática ha demostrado ser sumamente rica, aunque de gran complejidad respecto a su análisis. Dentro del conjunto de los sistemas no lineales, cabe destacar un reducido grupo, pero a la vez selecto, que se distingue por las propiedades extraordinarias que presenta: los denominados sistemas integrables. La presente tesis doctoral se centra en el estudio de algunas de las propiedades más relevantes observadas para los sistemas integrables. En esta tesis se pretende proporcionar un marco teórico unificado que permita abordar ecuaciones diferenciales no lineales que potencialmente puedan ser consideradas como integrables. En particular, el análisis de integralidad de dichas ecuaciones se realiza a través de técnicas basadas en la Propiedad de Painlevé, en combinación con la subsiguiente búsqueda de los problemas espectrales asociados y la identificación de soluciones analíticas de naturaleza solitónica. El método de la variedad singular junto con las transformaciones de auto-Bäcklund y de Darboux jugarán un papel fundamental en este estudio. Además, también se lleva a cabo un análisis complementario basado en las simetrías de Lie y reducciones de similaridad, que nos permitirán estudiar desde esta nueva perspectiva los problemas espectrales asociados. Partiendo de la archiconocida ecuación de Schrödinger no lineal, se han investigado diferentes generalizaciones integrables de dicha ecuación con numerosas aplicaciones en diversos campos científicos, como la Física Matemática, Física de Materiales o Biología.[EN] Nonlinear systems emerge as an active research topic of growing interest during the last decades due to their versatility when it comes to describing physical phenomena. Such scenarios are typically modelled by nonlinear differential equations, whose mathematical structure has proved to be incredibly rich, but highly nontrivial to treat. In particular, a narrow but surprisingly special group of this kind stands out: the so-called integrable systems. The present doctoral thesis focuses on the study of some of the extraordinary properties observed for integrable systems. The ultimate purpose of this dissertation lies in providing a unified theoretical framework that allows us to approach nonlinear differential equations that may potentially be considered as integrable. In particular, their integrability characterization is addressed by means of Painlevé analysis, in conjunction with the subsequent quest of the associated spectral problems and the identification of analytical solutions of solitonic nature. The singular manifold method together with auto-Bäckund and Darboux transformations play a critical role in this setting. In addition, a complementary methodology based on Lie symmetries and similarity reductions is proposed so as to analyze integrable systems by studying the symmetry properties of their associated spectral problems. Taking the ubiquitous nonlinear Schrödinger equation as the starting point, we have investigated several integrable generalizations of this equation that possess copious applications in distinct scientific fields, such as Mathematical Physics, Material Sciences and Biology

Geometric Analysis of Nonlinear Partial Differential Equations

This book contains a collection of twelve papers that reflect the state of the art of nonlinear differential equations in modern geometrical theory. It comprises miscellaneous topics of the local and nonlocal geometry of differential equations and the applications of the corresponding methods in hydrodynamics, symplectic geometry, optimal investment theory, etc. The contents will be useful for all the readers whose professional interests are related to nonlinear PDEs and differential geometry, both in theoretical and applied aspects

Integrable systems with BMS3_{3} Poisson structure and the dynamics of locally flat spacetimes

We construct a hierarchy of integrable systems whose Poisson structure corresponds to the BMS$_{3}$ algebra, and then discuss its description in terms of the Riemannian geometry of locally flat spacetimes in three dimensions. The analysis is performed in terms of two-dimensional gauge fields for $isl(2,R)$. Although the algebra is not semisimple, the formulation can be carried out \`a la Drinfeld-Sokolov because it admits a nondegenerate invariant bilinear metric. The hierarchy turns out to be bi-Hamiltonian, labeled by a nonnegative integer $k$, and defined through a suitable generalization of the Gelfand-Dikii polynomials. The symmetries of the hierarchy are explicitly found. For $k\geq 1$, the corresponding conserved charges span an infinite-dimensional Abelian algebra without central extensions, and they are in involution; while in the case of $k=0$, they generate the BMS$_{3}$ algebra. In the special case of $k=1$, by virtue of a suitable field redefinition and time scaling, the field equations are shown to be equivalent to a specific type of the Hirota-Satsuma coupled KdV systems. For $k\geq 1$, the hierarchy also includes the so-called perturbed KdV equations as a particular case. A wide class of analytic solutions is also explicitly constructed for a generic value of $k$. Remarkably, the dynamics can be fully geometrized so as to describe the evolution of spacelike surfaces embedded in locally flat spacetimes. Indeed, General Relativity in 3D can be endowed with a suitable set of boundary conditions, so that the Einstein equations precisely reduce to the ones of the hierarchy aforementioned. The symmetries of the integrable systems then arise as diffeomorphisms that preserve the asymptotic form of the spacetime metric, and therefore, they become Noetherian. The infinite set of conserved charges is recovered from the corresponding surface integrals in the canonical approach.Comment: 34 pages, 2 figure

Nonlinear classical and quantum integrable systems with PT -symmetries

A key feature of integrable systems is that they can be solved to obtain exact analytical solutions. In this thesis we show how new models can be found through generalisations of some well known nonlinear partial differential equations including the Korteweg-de Vries, modified Korteweg-de Vries, sine-Gordon, Hirota, Heisenberg and Landau-Lifschitz types with joint parity and time symmetries whilst preserving integrability properties. The first joint parity and time symmetric generalizations we take are extensions to the complex and multicomplex fields, such as bicomplex, quaternionic, coquaternionic and octonionic types. Subsequently, we develop new methods from well-known ones, such as Hirota’s direct method, Bäcklund transformations and Darboux-Crum transformations to solve for these newsystems to obtain exact analytical solutions of soliton and multi-soliton types. Moreover, in agreement with the reality property present in joint parity and time symmetric non-Hermitian quantum systems, we find joint parity and time symmetries also play a key role for reality of conserved charges for the new systems, even though the soliton solutions are complex or multicomplex. Our complex extensions have proved to be successful in helping one to obtain regularized degenerate multi-soliton solutions for the Korteweg-de Vries equation, which has not been realised before. We extend our investigations to explore degenerate multi-soliton solutions for the sine-Gordon equation and Hirota equation. In particular, we find the usual time-delays from degenerate soliton solution scattering are time-dependent, unlike the non-degenerate multi-soliton solutions, and provide a universal formula to compute the exact time-delay values for scattering of N-soliton solutions. Other joint parity and time symmetric extensions of integrable systems we take are of nonlocal nature, with nonlocalities in space and/or in time, of time crystal type. Whilst developing new methods for the construction of soliton solutions for these systems, we xiv find new types of solutions with different parameter dependence and qualitative behaviour even in the one-soliton solution cases. We exploit gauge equivalence between the Hirota system with continuous Heisenberg and Landau-Lifschitz systems to see how nonlocality is inherited from one system to another and vice versa. In the final part of the thesis, we extend some of our investigations to the quantum regime. In particularwe generalize the scheme of Darboux transformations for fully timedependent non-Hermitian quantum systems, which allows us to create an infinite tower of solvable models

The higher grading structure of the WKI hierarchy and the two-component short pulse equation

A higher grading affine algebraic construction of integrable hierarchies, containing the Wadati-Konno-Ichikawa (WKI) hierarchy as a particular case, is proposed. We show that a two-component generalization of the Sch\" afer-Wayne short pulse equation arises quite naturally from the first negative flow of the WKI hierarchy. Some novel integrable nonautonomous models are also proposed. The conserved charges, both local and nonlocal, are obtained from the Riccati form of the spectral problem. The loop-soliton solutions of the WKI hierarchy are systematically constructed through gauge followed by reciprocal B\" acklund transformation, establishing the precise connection between the whole WKI and AKNS hierarchies. The connection between the short pulse equation with the sine-Gordon model is extended to a correspondence between the two-component short pulse equation and the Lund-Regge model

A third-order nonlinear Schrodinger equation: The exact solutions, group-invariant solutions and conservation laws

In this study, we consider the third order nonlinear Schrodinger equation (TONSE) that models the wave pulse transmission in a time period less than one-trillionth of a second. With the help of the extended modified method, we obtain numerous exact travelling wave solutions containing sets of generalized hyperbolic, trigonometric and rational solutions that are more general than classical ones. Secondly, we construct the transformation groups which left the equations invariant and vector fields with the Lie symmetry groups approach. With the help of these vector fields, we obtain the symmetry reductions and exact solutions of the equation. The obtained group-invariant solutions are Jacobi elliptic function and exponential type. We discuss the dynamic behaviour and structure of the exact solutions for distinct solutions of arbitrary constants. Lastly, we obtain conservation laws of the considered equation by construing the complex equation as a system of two real partial differential equations (PDEs)

Nonlinear wave patterns in the complex KdV and nonlinear Schrodinger equations

This thesis is on the theory of nonlinear waves in physics. To begin with, we develop from first principles the theory of the complex Korteweg-de Vries (KdV) equation as an equation for the complex velocity of a weakly nonlinear wave in a shallow, ideal fluid. We show that this is completely consistent with the well-known theory of the real KdV equation as a special case, but has the advantage of directly giving complete information about the motion of all particles within the fluid. We show that the complex KdV equation also has conserved quantities which are completely consistent with the physical interpretation of the real KdV equation. When a periodic wave solution to the real KdV equation is expanded in the quasi-monochromatic approximation, it is known that the amplitude of the wave envelope is described by the nonlinear Schrodinger (NLS) equation. However, in the complex KdV equation, we show that the fundamental modes of the velocity are described by the split NLS equations, themselves a special case of the Ablowitz-Kaup-Newell-Segur system. This is a directly physical interpretation of the split NLS equations, which were primarily introduced as only a mathematical construct emerging from the Zakharov-Shabat equations. We also discuss an empirically obtained symmetry of the rational solutions to the KdV equations, which seems to have been unnoticed until now. Solutions which can be written in terms of Wronskian determinants are well-known; however, we show that these are actually part of a more general family of rational solutions. We show that a linear combination of the Wronskians of orders $n$ and $n+2$ generates a new, multi-peak rational solution to the KdV equation. We next move on to the integrable extensions of the NLS equation. These incorporate higher order nonlinear and dispersive terms in such a way that the system keeps the same conserved quantities, and is thus completely integrable. We obtain the general solution of the doubly-periodic solutions of the class I extension of the NLS equation, and discuss several special cases. These are the most general one-parameter first order solutions of the (class I) extended NLS equation. Building on this, we also discuss second order solutions to the extended NLS equation. We obtain the general 2-breather solutions, and discuss several special cases; among them, semirational breathers, the degenerate breather solution, the second-order rogue wave, and the rogue wave triplet solution. We also discuss the breather to soliton conversion, which is a solution which does not exist in the basic NLS equation where only the lowest order dispersive and nonlinear terms are present. Finally, we discuss a few possibilities for future research based on the work done in this thesis