The design of conservative finite element discretisations for the vectorial modified KdV equation

We design a consistent Galerkin scheme for the approximation of the vectorial modified Korteweg-de Vries equation. We demonstrate that the scheme conserves energy up to machine precision. In this sense the method is consistent with the energy balance of the continuous system. This energy balance ensures there is no numerical dissipation allowing for extremely accurate long time simulations free from numerical artifacts. Various numerical experiments are shown demonstrating the asymptotic convergence of the method with respect to the discretisation parameters. Some simulations are also presented that correctly capture the unusual interactions between solitons in the vectorial setting

Generalization of nonlocally related partial differential equation systems: unknown symmetric properties and analytical solutions

Symmetry, which describes invariance, is an eternal concern in mathematics and physics, especially in the investigation of solutions to the partial differential equation (PDE). A PDE's nonlocally related PDE systems provide excellent approaches to search for various symmetries that expand the range of its known solutions. They composed of potential systems based on conservation laws and inverse potential systems (IPS) based on differential invariants. Our study is devoted to generalizing their construction and application in three-dimensional circumstances. Concretely, the potential of the algebraic gauge-constrained potential system is simplified without weakening its solution space. The potential system is extended via nonlocal conservation laws and double reductions. Afterwards, nonlocal symmetries are identified in the IPS.\@ The IPS is extended by the solvable Lie algebra and type \Rmnum{2} hidden symmetries. Besides, systems among equations can be connected via Cole-Hopf transformation.\@ Ultimately, established and extended systems embody rich symmetric properties and unprecedented analytical solutions, and may even further facilitate general coordinate-independent analysis in qualitative, numerical, perturbation, etc., this can be illustrated by several Burgers-type equations

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