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Abstract

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.

Publication Type:	Thesis (Doctoral)
Subjects:	Q Science > QA Mathematics
Departments:	Doctoral Theses School of Science & Technology > School of Science & Technology Doctoral Theses School of Science & Technology > Mathematics

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