38 research outputs found
Bipolar solitary wave interactions within the Schamel equation
Pair soliton interactions play a significant role in the dynamics of soliton
turbulence. The interaction of solitons with different polarities is
particularly crucial in the context of abnormally large wave formation, often
referred to as freak or rogue waves, as these interactions result in an
increase in the maximum wave field. In this article, we investigate the
features and properties of bipolar soliton interactions within the framework of
the non-integrable Schamel equation, contrasting them with the integrable
modified Korteweg-de Vries equation. We examine variations in moments and
extrema of the wave fields. Additionally, we identify scenarios in which, in
the bipolar solitary wave interaction, the smaller solitary wave transfers a
portion of its energy to the larger one, causing an increase in the amplitude
of the larger solitary wave and a decrease in the amplitude of the smaller one,
returning them to their pre-interaction state. Notably, we observe that
non-integrability can be considered a factor that triggers the formation of
rogue waves
A universal asymptotic regime in the hyperbolic nonlinear Schr\"odinger equation
The appearance of a fundamental long-time asymptotic regime in the two space
one time dimensional hyperbolic nonlinear Schr\"odinger (HNLS) equation is
discussed. Based on analytical and extensive numerical simulations an
approximate self-similar solution is found for a wide range of initial
conditions -- essentially for initial lumps of small to moderate energy. Even
relatively large initial amplitudes, which imply strong nonlinear effects,
eventually lead to local structures resembling those of the self-similar
solution, with appropriate small modifications. These modifications are
important in order to properly capture the behavior of the phase of the
solution. This solution has aspects that suggest it is a universal attractor
emanating from wide ranges of initial data.Comment: 36 pages, 26 pages text + 20 figure
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Nonlinear Waves and Dispersive Equations
Nonlinear dispersive equations are models for nonlinear waves in a wide range of physical contexts. Mathematically they display an interplay between linear dispersion and nonlinear interactions, which can result in a wide range of outcomes from finite time blow-up to scattering. They are linked to many areas of mathematics and physics, ranging from integrable systems and harmonic analysis to fluid dynamics and general relativity. The conference did focus on the analytic aspects and PDE aspects
Beyond the KdV: post-explosion development
Several threads of the last 25 years’ developments in nonlinear wave theory that stem from the classical Korteweg–de Vries (KdV) equation are surveyed. The focus is on various generalizations of the KdV equation which include higher-order nonlinearity, large-scale dispersion, and a nonlocal integral dispersion. We also discuss how relatively simple models can capture strongly nonlinear dynamics and how various modifications of the KdV equation lead to qualitatively new, non-trivial solutions and regimes of evolution observable in the laboratory and in nature. As the main physical example, we choose internal gravity waves in the ocean for which all these models are applicable and have genuine importance. We also briefly outline the authors’ view of the future development of the chosen lines of nonlinear wave theory
Dispersive Shock Wave, Generalized Laguerre Polynomials and Asymptotic Solitons of the Focusing Nonlinear Schr\"odinger Equation
We consider dispersive shock wave to the focusing nonlinear Schr\"odinger
equation generated by a discontinuous initial condition which is periodic or
quasi-periodic on the left semi-axis and zero on the right semi-axis. As an
initial function we use a finite-gap potential of the Dirac operator given in
an explicit form through hyper-elliptic theta-functions. The paper aim is to
study the long-time asymptotics of the solution of this problem in a vicinity
of the leading edge, where a train of asymptotic solitons are generated. Such a
problem was studied in \cite{KK86} and \cite{K91} using Marchenko's inverse
scattering technics. We investigate this problem exceptionally using the
Riemann-Hilbert problems technics that allow us to obtain explicit formulas for
the asymptotic solitons themselves that in contrast with the cited papers where
asymptotic formulas are obtained only for the square of absolute value of
solution. Using transformations of the main RH problems we arrive to a model
problem corresponding to the parametrix at the end points of continuous
spectrum of the Zakharov-Shabat spectral problem. The parametrix problem is
effectively solved in terms of the generalized Laguerre polynomials which are
naturally appeared after appropriate scaling of the Riemann-Hilbert problem in
a small neighborhoods of the end points of continuous spectrum. Further
asymptotic analysis give an explicit formula for solitons at the edge of
dispersive wave. Thus, we give the complete description of the train of
asymptotic solitons: not only bearing envelope of each asymptotic soliton, but
its oscillating structure are found explicitly. Besides the second term of
asymptotics describing an interaction between these solitons and oscillating
background is also found. This gives the fine structure of the edge of
dispersive shock wave.Comment: 36 pages, 5 figure
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