Breaking of Symmetrical Periodic Solutions In A Singularly Perturbed Kdv Model

There are several recent developments in the well-known problem of breaking of homoclinic orbits (splitting of separatrices) of a system that undergoes a singular perturbation. First, survival of a homoclinic orbit is an exceptional situation that can be linked to triviality of the Stokes phenomenon of the underlying truncated equation. Second, homoclinic connections to exponentially small periodic orbits survive the perturbation in the generic case. In this paper we consider a different problem: we study deformations of genuine periodic orbits of the second order equation y \u27\u27 = y + y(2) that undergoes the singular perturbation epsilon(2)y \u27\u27\u27\u27 + (1 - epsilon(2))y \u27\u27 = y + y(2), where epsilon \u3e 0 is a small parameter. We prove that if the period and the constant of motion do not change too rapidly (in epsilon), a genuine (nontrivial) periodic solution does not survive the perturbation

Semiclassical limit of the focusing NLS: Whitham equations and the Riemann-Hilbert Problem approach

The main goal of this paper is to put together: a) the Whitham theory applicable to slowly modulated N-phase nonlinear wave solutions to the focusing nonlinear Schr odinger (fNLS) equation, and b) the Riemann- Hilbert Problem approach to particular solutions of the fNLS in the semiclassical (small dispersion) limit that develop slowly modulated N-phase nonlinear wave in the process of evolution. Both approaches have their own merits and limitations. Understanding of the interrelations between them could prove benefi cial for a broad range of problems involving the semiclassical fNLS

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

Structure and dynamics of solitary waves in fluid media

This research deals with the study of nonlinear solitary waves in fluid media. The equations which model surface and internal waves in fluids have been studied and used in this research. The approach to study the structure and dynamics of internal solitary waves in near-critical situations is the traditional theoretical and numerical study of nonlinear wave processes based on the methods of dynamical systems. The synergetic approach has been exploited, which presumes a combination of theoretical and numerical methods. All numerical calculations were performed with the desktop personal computer. Traditional and novel methods of mathematical physics were actively used, including Fourier analysis technique, inverse scattering method, Hirota method, phase-plane analysis, analysis of integral invariants, finite-difference method, Petviashvili and Yang–Lakoba numerical iterative techniques for the numerical solution of Partial Differential Equation. A new model equation, dubbed the Gardner–Kawahara equation, has been suggested to describe wave phenomena in the near-critical situations, when the nonlinear and dispersive coefficients become anomalously small. Such near-critical situations were not studied so far, therefore this study is very topical and innovative. Results obtained will shed a light on the structure of solitary waves in near-critical situation, which can occur in two-layer fluid with strong surface tension between the layers. A family of solitary waves was constructed numerically for the derived Gardner–Kawahara equation; their structure has been investigated analytically and numerically. The problem of modulation stability of quasi-monochromatic wave-trains propagating in a media has also being studied. The Nonlinear Schrödinger equation (NLSE) has been derived from the unidirectional Gardner–Ostrovsky equation and a general Shrira equation which describes both surface and internal long waves in a rotating fluid. It was demonstrated that earlier obtained results (Grimshaw & Helfrich, 2008; 2012; Whitfield & Johnson, 2015a; 2015b) on modulational stability/instability are correct within the limited range of wavenumbers where the Ostrovsky equation is applicable. In the meantime, results obtained in this Thesis and published in the paper (Nikitenkova et al., 2015) are applicable in the wider range of wavenumbers up to k = 0. It was shown that surface and internal oceanic waves are stable with respect to selfmodulation at small wavenumbers when k → 0 in contrast to what was mistakenly obtained in (Shrira, 1981). In Chapter 4 new exact solutions of the Kadomtsev-Petviashvili equation with a positive dispersion are obtained in the form of obliquely propagating skew lumps. Specific features of such lumps were studied in details. In particular, the integral characteristics of single lumps (mass, momentum components and energy) have been calculated and presented in terms of lump velocity. It was shown that exact stationary multi-lump solutions can be constructed for this equation. As the example, the exact bilump solution is presented in the explicit form and illustrated graphically. The relevance of skew lumps to the real physical systems is discussed

14th Conference on Dynamical Systems Theory and Applications DSTA 2017 ABSTRACTS

From Preface: This is the fourteen time when the conference “Dynamical Systems – Theory and Applications” gathers a numerous group of outstanding scientists and engineers, who deal with widely understood problems of theoretical and applied dynamics. Organization of the conference would not have been possible without a great effort of the staff of the Department of Automation, Biomechanics and Mechatronics. The patronage over the conference has been taken by the Committee of Mechanics of the Polish Academy of Sciences and the Ministry of Science and Higher Education. It is a great pleasure that our invitation has been accepted by so many people, including good colleagues and friends as well as a large group of researchers and scientists, who decided to participate in the conference for the first time. With proud and satisfaction we welcome nearly 250 persons from 38 countries all over the world. They decided to share the results of their research and many years experiences in the discipline of dynamical systems by submitting many very interesting papers. This booklet contains a collection of 375 abstracts, which have gained the acceptance of referees and have been qualified for publication in the conference proceedings [...]

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