Nonlinear Evolution Equations: Analysis and Numerics

The qualitative theory of nonlinear evolution equations is an important tool for studying the dynamical behavior of systems in science and technology. A thorough understanding of the complex behavior of such systems requires detailed analytical and numerical investigations of the underlying partial differential equations

Symmetry and double reduction for exact solutions of selected nonlinear partial differential equations

Amongst the several analytic methods available to obtain exact solutions of non-linear differential equations, Lie symmetry reduction and double reduction technique are proven to be most effective and have attracted researcher from different areas to utilize these methods in their research. In this research, Lie symmetry analysis and double reduction are used to find the exact solutions of nonlinear differential equations. For Lie symmetry reduction method, symmetries of differential equation will be obtained and hence invariants will be obtained, thus differential equation will be reduced and exact solutions are calculated. For the method of double reduction, we first find Lie symmetry, followed by conservation laws using ‘Multiplier’ approach. Finally, possibilities of associations between symmetry with conservation law will be used to reduce the differential equation, and thereby solve the differential equation. These methods will be used on some physically very important nonlinear differential equations; such as Kadomtsev- Petviashvili equation, Boyer-Finley equation, Short Pulse Equation, and Kortewegde Vries-Burgers equations. Furthermore, verification of the solution obtained also will be done by function of PDETest integrated in Maple or comparison to exist literature

Soliton solutions to the fifth-order Korteweg–de Vries equation and their applications to surface and internal water waves

We study solitary wave solutions of the fifth-order Korteweg–de Vries equation which contains, besides the traditional quadratic nonlinearity and third-order dispersion, additional terms including cubic nonlinearity and fifth order linear dispersion, as well as two nonlinear dispersive terms. An exact solitary wave solution to this equation is derived, and the dependence of its amplitude, width, and speed on the parameters of the governing equation is studied. It is shown that the derived solution can represent either an embedded or regular soliton depending on the equation parameters. The nonlinear dispersive terms can drastically influence the existence of solitary waves, their nature (regular or embedded), profile, polarity, and stability with respect to small perturbations. We show, in particular, that in some cases embedded solitons can be stable even with respect to interactions with regular solitons. The results obtained are applicable to surface and internal waves in fluids, as well as to waves in other media (plasma, solid waveguides, elastic media with microstructure, etc.)

Nonlinear Evolution Equations: Analysis and Numerics

The workshop was devoted to the analytical and numerical investigation of nonlinear evolution equations. The main aim was to stimulate a closer interaction between experts in analytical and numerical methods for areas such as wave and Schrödinger equations or the Navier–Stokes equations and fluid dynamics

Numerical Analysis and Theory of Oblique Alfvenic Solitons Observed in the Interplanetary Magnetic Field

Recently, there have been reports of small magnetic pulses or bumps in the interplanetary magnetic field observed by various spacecraft. Most of these reports claim that these localized pulses or bumps are solitons. Solitons are weakly nonlinear localized waves that tend to retain their form as they propagate and can be observed in various media which exhibit nonlinear steepening and dispersive effects. This thesis expands the claim that these pulses or bumps are nonlinear oblique Alfven waves with soliton components, through the application of analytical techniques used in the inverse scattering transform in a numerical context and numerical integration of nonlinear partial dierential equations. One event, which was observed by the Ulysses spacecraft on February 21st, 2001, is extensively scrutinized through comparison with soliton solutions that emerge from the Derivative Nonlinear Schrodinger (DNLS) equation. The direct scattering transform of a wave prole that has corresponding morphology to the selected magnetic bump leads to the implication of a soliton component. Numerical integration of the scaled prole matching the event in the context of the DNLS leads to generation of dispersive waves and a one parameter dark soliton

Multiscale multiphysics data-informed modeling for three-dimensional ocean acoustic simulation and prediction

Author Posting. © Acoustical Society of America, 2019. This article is posted here by permission of Acoustical Society of America for personal use, not for redistribution. The definitive version was published in Journal of the Acoustical Society of America 146(3), (2019): 1996-2015, doi:10.1121/1.5126012.Three-dimensional (3D) underwater sound field computations have been used for a few decades to understand sound propagation effects above sloped seabeds and in areas with strong 3D temperature and salinity variations. For an approximate simulation of effects in nature, the necessary 3D sound-speed field can be made from snapshots of temperature and salinity from an operational data-driven regional ocean model. However, these models invariably have resolution constraints and physics approximations that exclude features that can have strong effects on acoustics, example features being strong submesoscale fronts and nonhydrostatic nonlinear internal waves (NNIWs). Here, work to predict NNIW fields to improve 3D acoustic forecasts using an NNIW model nested in a tide-inclusive data-assimilating regional model is reported. The work was initiated under the Integrated Ocean Dynamics and Acoustics project. The project investigated ocean dynamical processes that affect important details of sound-propagation, with a focus on those with strong intermittency (high kurtosis) that are challenging to predict deterministically. Strong internal tides and NNIW are two such phenomena, with the former being precursors to NNIW, often feeding energy to them. Successful aspects of the modeling are reported along with weaknesses and unresolved issues identified in the course of the work.This work was supported by Department of Defense Multidisciplinary University Initiative (MURI) Grant No. N00014-11-1-0701, managed by the Office of Naval Research Ocean Acoustics Program, and National Science Foundation Grant No. OCE-1060430. Final manuscript preparation was supported by ONR Ocean Acoustics Grant Nos. N00014-17-1-2624 and N00014-17-1-2692. P.F.J.L. also thanks ONR and NSF for research support under Grant Nos. N00014-13-1-0518 (Multi-DA) and OCE-1061160 (ShelfIT) to MIT, respectively. The MSEAS-based series of simulations for the New Jersey shelf region examined here was accelerated toward completion by the interest in realistic 3D acoustic fields expressed by Dr. Ivars Kirsteins at the Naval Undersea Warfare Center.2020-03-3