Transformation of a shoaling undular bore

We consider the propagation of a shallow-water undular bore over a gentle monotonic bottom slope connecting two regions of constant depth, in the framework of the variable-coefficient Korteweg-de Vries equation. We show that, when the undular bore advances in the direction of decreasing depth, its interaction with the slowly varying topography results, apart from an adiabatic deformation of the bore itself, in the generation of a sequence of isolated solitons - an expanding large-amplitude modulated solitary wavetrain propagating ahead of the bore. Using nonlinear modulation theory we construct an asymptotic solution describing the formation and evolution of this solitary wavetrain. Our analytical solution is supported by direct numerical simulations. The presented analysis can be extended to other systems describing the propagation of undular bores (dispersive shock waves) in weakly non-uniform environments

A novel implementation of Petrov-Galerkin method to shallow water solitary wave pattern and superperiodic traveling wave and its multistability: generalized Korteweg-de Vries equation

This work deals with the constitute of numerical solutions of the generalized Korteweg-de Vries (GKdV) equation with Petrov-Galerkin finite element approach utilising a cubic B-spline function as the trial function and a quadratic function as the test function. Accurateness and effectiveness of the submitted methods are shown by employing propagation of single solitary wave. The L2, L∞error norms and I1, I2and I3invariants are used to validate the applicability and durability of our numerical algorithm. Implementing the Von-Neumann theory, it is manifested that the suggested method is marginally stable. Furthermore, supernonlinear traveling wave solution of the GKdV equation is presented using phase plots. It is seen that the GKdV equation supports superperiodic traveling wave solution only and it is significantly affected by velocity and nonlinear parameters. Also, considering a superficial periodic forcing multistability of traveling waves of perturbed GKdV equation is presented. It is found that the perturbed GKdV equation supports coexisting chaotic and various quasiperiodic features with same parametric values at different initial condition

Dispersive shock waves and modulation theory

There is growing physical and mathematical interest in the hydrodynamics of dissipationless/dispersive media. Since G. B. Whitham’s seminal publication fifty years ago that ushered in the mathematical study of dispersive hydrodynamics, there has been a significant body of work in this area. However, there has been no comprehensive survey of the field of dispersive hydrodynamics. Utilizing Whitham’s averaging theory as the primary mathematical tool, we review the rich mathematical developments over the past fifty years with an emphasis on physical applications. The fundamental, large scale, coherent excitation in dispersive hydrodynamic systems is an expanding, oscillatory dispersive shock wave or DSW. Both the macroscopic and microscopic properties of DSWs are analyzed in detail within the context of the universal, integrable, and foundational models for uni-directional (Korteweg–de Vries equation) and bi-directional (Nonlinear Schrödinger equation) dispersive hydrodynamics. A DSW fitting procedure that does not rely upon integrable structure yet reveals important macroscopic DSW properties is described. DSW theory is then applied to a number of physical applications: superfluids, nonlinear optics, geophysics, and fluid dynamics. Finally, we survey some of the more recent developments including non-classical DSWs, DSW interactions, DSWs in perturbed and inhomogeneous environments, and two-dimensional, oblique DSWs

Differential/Difference Equations

The study of oscillatory phenomena is an important part of the theory of differential equations. Oscillations naturally occur in virtually every area of applied science including, e.g., mechanics, electrical, radio engineering, and vibrotechnics. This Special Issue includes 19 high-quality papers with original research results in theoretical research, and recent progress in the study of applied problems in science and technology. This Special Issue brought together mathematicians with physicists, engineers, as well as other scientists. Topics covered in this issue: Oscillation theory; Differential/difference equations; Partial differential equations; Dynamical systems; Fractional calculus; Delays; Mathematical modeling and oscillations

New Advancements in Pure and Applied Mathematics via Fractals and Fractional Calculus

This reprint focuses on exploring new developments in both pure and applied mathematics as a result of fractional behaviour. It covers the range of ongoing activities in the context of fractional calculus by offering alternate viewpoints, workable solutions, new derivatives, and methods to solve real-world problems. It is impossible to deny that fractional behaviour exists in nature. Any phenomenon that has a pulse, rhythm, or pattern appears to be a fractal. The 17 papers that were published and are part of this volume provide credence to that claim. A variety of topics illustrate the use of fractional calculus in a range of disciplines and offer sufficient coverage to pique every reader's attention

New exact solutionsand numerical approximations of the generalized kdv equation

This paper is devoted to create new exact and numerical solutions of the generalized Korteweg-de Vries (GKdV) equation with ansatz method and Galerkin finite element method based on cubic B-splines over finite elements. Propagation of single solitary wave is investigated to show the efficiency and applicability of the proposed methods. The performance of the numerical algorithm is proved by computing L2 and L∞ error norms. Also, three invariants I1, I2, and I3 have been calculated to determine the conservation properties of the presented algorithm. The obtained numerical solutions are compared with some earlier studies for similar parameters. This comparison clearly shows that the obtained results are better than some earlier results and they are found to be in good agreement with exact solutions. Additionally, a linear stability analysis based on Von Neumann’s theory is surveyed and indicated that our method is unconditionally stable

Applied Mathematics and Computational Physics

As faster and more efficient numerical algorithms become available, the understanding of the physics and the mathematical foundation behind these new methods will play an increasingly important role. This Special Issue provides a platform for researchers from both academia and industry to present their novel computational methods that have engineering and physics applications

2009 program of studies : nonlinear waves

The fiftieth year of the program was dedicated to Nonlinear Waves, a topic with many applications in geophysical fluid dynamics. The principal lectures were given jointly by Roger Grimshaw and Harvey Segur and between them they covered material drawn from fundamental theory, fluid experiments, asymptotics, and reaching all the way to detailed applications. These lectures set the scene for the rest of the summer, with subsequent daily lectures by staff and visitors on a wide range of topics in GFD. It was a challenge for the fellows and lecturers to provide a consistent set of lecture notes for such a wide-ranging lecture course, but not least due to the valiant efforts of Pascale Garaud, who coordinated the write-up and proof-read all the notes, we are very pleased with the final outcome contained in these pages. This year’s group of eleven international GFD fellows was as diverse as one could get in terms of gender, origin, and race, but all were unified in their desire to apply their fundamental knowledge of fluid dynamics to challenging problems in the real world. Their projects covered a huge range of physical topics and at the end of the summer each student presented his or her work in a one-hour lecture. As always, these projects are the heart of the research and education aspects of our summer study.Funding was provided by the National Science Foundation through Grant No. OCE-0824636 and the Office of Naval Research under Contract No. N00014-09-10844

Dispersive hydrodynamics in a non-local non-linear medium

Dispersive shock wave (DSW), sometimes referred to as an undular bore in fluid mechanics, is a non-linear dispersive wave phenomenon which arises in non-linear dispersive media for which viscosity effects are negligible or non-existent. It is generated when physical quantities, such as fluid pressure, density, temperature and electromagnetic wave intensity, undergo rapid variations as time evolves. Its structure is a non-stationary modulated wavetrain which links two distinct physical states. DSW's occurrence in nature is quite omnipresent in classical/quantum fluids and non-linear optics. The main purpose of this thesis is to fully analyse all regimes for DSW propagation in the non-linear optical medium of a nematic liquid crystal in the defocusing regime. These DSWs are generated from step initial conditions for the intensity of the optical field and are resonant in that linear diffractive waves (termed dispersive waves in the context of fluid mechanics) are in resonance with the DSW, leading to a resonant wavetrain propagating ahead of it. It is found that there are six hydrodynamic regimes, which are distinct and require different solution methods. In previous studies, a reductive nematic Korteweg-de Vries equation and gas dynamic shock wave theory were used to understand all nematic dispersive hydrodynamics, which do not yield solutions in full agreement with numerical solutions. Indeed, the standard DSW structure disappears and a ``Whitham shock'' emerges for sufficiently large initial jumps. Asymptotic theory, approximate methods or Whitham's modulation theory are used to find solutions for these resonant DSWs in a given regime. It is found that for small initial intensity jumps, the resonant wavetrain is unstable, but that it stabilises above a critical jump height. It is additionally found that the DSW is unstable, except for small jump heights for which there is no resonance and large jump heights for which there is no standard DSW structure. The theoretical solutions are found to be in excellent agreement with numerical solutions of the nematic equations in all hydrodynamic regimes