A splitting lattice Boltzmann scheme for (2+1)-dimensional soliton solutions of the Kadomtsev-Petviashvili equation

Recently, considerable attention has been given to (2+1)-dimensional Kadomtsev-Petviashvili equations due to their extensive applications in solitons that widely exist in nonlinear science. Therefore, developing a reliable numerical algorithm for the Kadomtsev-Petviashvili equations is crucial. The lattice Boltzmann method, which has been an efficient simulation method in the last three decades, is a promising technique for solving Kadomtsev-Petviashvili equations. However, the traditional higher-order moment lattice Boltzmann model for the Kadomtsev-Petviashvili equations suffers from low accuracy because of error accumulation. To overcome this shortcoming, a splitting lattice Boltzmann scheme for (2+1)-dimensional Kadomtsev-Petviashvili-Ⅰ type equations is proposed in this paper. The variable substitution method is applied to transform the Kadomtsev-Petviashvili-Ⅰ type equation into two macroscopic equations. Two sets of distribution functions are employed to construct these two macroscopic equations. Moreover, three types of soliton solutions are numerically simulated by this algorithm. The numerical results imply that the splitting lattice Boltzmann schemes have an advantage over the traditional high-order moment lattice Boltzmann model in simulating the Kadomtsev-Petviashvili-Ⅰ type equations

Hydrodynamic scales of integrable many-particle systems

1. Introduction, 2. Dynamics of the classical Toda lattice, 3. Static properties, 4. Dyson Brownian motion. , 5. Hydrodynamics for hard rods, 6. Equations of generalized hydrodynamics, 7. Linearized hydrodynamics and GGE dynamical correlations, 8. Domain wall initial states, 9. Toda fluid, 10. Hydrodynamics of soliton gases, 11. Calogero models, 12. Discretized nonlinear Schr\"odinger equation , 13. Hydrodynamics for the Lieb-Liniger $\delta$-Bose gas, 14. Quantum Toda lattice, 15. Beyond the Euler time scaleComment: 178 pages, 12 Figures. This a much enlarged and substantially improved version of arXiv:2101.0652

Minimal superstring theories

In this thesis we study the (2, 4m) series of Type OA minimal superstring theories, which are intimately related to the KdV integrable hierarchy. Chapters 1 to 3 constitute a comprehensive review of the relevant background material, including noncritical string theory, matrix models and integrable systems. In Chapter 4 we generalise the (2， 4m) theories so as to include unoriented worldsheet contributions in the partition sum. This generalisation is tested against a known result in the literature. Chapter 5 explores the D-branes of the theories in more detail and, by studying the Bäcklund transformation of the KdV hierarchy, gives conclusive evidence that the parameter speculated to control the number of zz branes in the theory is indeed quantised. The FZZT brane partition function is studied, and it is shown that the effects of the boundary cosmological constant arise only in certain predictable forms. Chapter 6 examines the string theory interpretation of the negative KdV hierarchy, which naively relates to supercritical string theories living in greater than ten dimensions. The models are seen to have some non-trivial characteristic properties, indicating that they do indeed describe valid string theories, but it is unclear whether said theories are actually supercritical

Integrable systems, random matrices and applications

Nonlinear integrable systems emerge in a broad class of different problems in Mathematics and Physics. One of the most relevant characterisation of integrable systems is the existence of an infinite number of conservation laws, associated to integrable hierarchies of equations. When nonlinearity is involved, critical phenomena may occur. A solution to a nonlinear partial differential equation may develop a gradient catastrophe and the consequent formation of a shock at the critical point. The approach of differential identities provides a convenient description of systems affected by phase transitions, identifying a suitable nonlinear equation for the order parameter of the system. This thesis is aimed to give a contribution to the perspective offered by the approach of differential identities. We discuss how this method is particularly useful in treating mean-field theories, with some explicit application. The core of the work concerns the Hermitian matrix ensemble and the symmetric matrix ensemble, analysed in the context of integrable systems. They both underlie a discrete integrable structure in form of a lattice, satisfying a discrete integrable hierarchy. We have studied a particular reduction of both system and determined the continuum limit of the dynamics of the field variables at the leading order. Particular emphasis has been given to the study of the symmetric matrix ensemble. We have unveiled an unobserved double-chain structure shared by the field variables populating the lattice structure associated to the ensemble. In the continuum limit of a particular reduction of the lattice, we have found a new hydrodynamic chain, a hydrodynamic system with infinitely many components. We have shown that the hydrodynamic chain is integrable and we have conjectured the form of the associated hierarchy. The new integrable hydrodynamic chain constitutes per se an interesting object of study. Indeed, it presents some properties that are different from those shared by the standard integrable hydrodynamic chains studied in literature

1980 summer study program in geophysical fluid dynamics : coherent features in geophysical flows

Four principal lecturers shored the task of presenting the subject "Coherent Features in Geophysical Flows" to the participants of the twenty-second geophysical fluid dynamics summer program. Glenn Flierl introduced the topic and the Kortweg-de Vries equation via a model of finite amplitude motions on the beta plane. He extended the analysis to more complex flows in the ocean and the atmosphere and in the process treated motions of very large amplitude. Larry Redekopp's three lectures summarized an extensive body of the mathematical literature on coherent features. Andrew Ingersoll focussed on the many fascinating features in Jupiter's atmosphere. Joseph Keller supplemented an interesting summary of laboratory observations with suggestive models for treating the flows.Office of Naval Research under Contract N00014-79-C-067

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described