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Anna Karczewska

Publications and source records attributed to Anna Karczewska.

At least 37 records · Page 2Linked to original sources
arXiv2017

Exact solitonic and periodic solutions of the extended KdV equation

The KdV equation can be derived in the shallow water limit of the Euler equations. Over the last few decades, this equation has been extended to include both higher order effects (KdV2) and an uneven river bottom. Although this equation is not integrable and has only one conservation law, exact periodic and solitonic solutions exist for the even bottom case. The method used to find them assumes the same function forms as for KdV solutions. KdV2 equation imposes more constraints on parameters of solutions. For soliton case KdV2 solution occurs for particular ratio of wave amplitude to water depth only. For periodic case physically relevant solutions are admissible only for two narrow intervals of elliptic parameter $m$. For a range of $m$ near one the cnoidal waves are upright as expected, but are inverted in $m$ region close to zero. Properties of exact solutions of KdV and KdV2 are compared.

physics.flu-dyn↗
arXiv2016

Adiabatic invariants of the extended KdV equation

When the Euler equations for shallow water are taken to the next order, beyond KdV, momentum and energy are no longer exact invariants. (The only one is mass.) However, adiabatic invariants (AI) can be found. When the KdV expansion parameters are zero, exact invariants are recovered. Existence of adiabatic invariants results from general theory of near-identity transformations (NIT) which allow us to transform higher order nonintegrable equations to asymptotically equivalent (when small parameters tend to zero) integrable form. Here we present the direct method of calculations of adiabatic invariants. It does not need a transformation to a moving reference frame nor performing a near-identity transformation. Numerical tests show that deviations of AI from almost constant values are indeed small.

physics.flu-dyn↗
arXiv2016

Finite Element Method for Stochastic Extended KdV Equations

The finite element method is applied to obtain numerical solutions to the recently derived nonlinear equation for shallow water wave problem for several cases of bottom shapes. Results for time evolution of KdV solitons and cnoidal waves under stochastic forces are presented. Though small effects originating from second order dynamics may be obscured by stochastic forces, the main waves, both cnoidal and solitary ones, remain very robust against any distortions.

physics.flu-dyn↗
arXiv2016

Finite element method for extended KdV equations

The finite element method (FEM) is applied to obtain numerical solutions to a recently derived nonlinear equation for the shallow water wave problem. A weak formulation and the Petrov-Galerkin method are used. It is shown that the FEM gives a reasonable description of the wave dynamics of soliton waves governed by extended KdV equations. Some new results for several cases of bottom shapes are presented. The numerical scheme presented here is suitable for taking into account stochastic effects, which will be discussed in a subsequent paper.

physics.flu-dyn↗
arXiv2015

Energy invariant for shallow water waves and the Korteweg -- de Vries equation. Is energy always an invariant?

It is well known that the KdV equation has an infinite set of conserved quantities. The first three are often considered to represent mass, momentum and energy. Here we try to answer the question of how this comes about, and also how these KdV quantities relate to those of the Euler shallow water equations. Here Luke's Lagrangian is helpful. We also consider higher order extensions of KdV. Though in general not integrable, in some sense they are almost so.

physics.flu-dyn↗
arXiv2014

Shallow water soliton dynamics beyond KdV

An alternative way for the derivation of the new KdV-type equation is presented. The equation contains terms depending on the bottom topography (there are six new terms in all, three of which are caused by the unevenness of the bottom). It is obtained in the second order perturbative approach in the weakly nonlinear, dispersive and long wavelength limit. Only treating all these terms in the second order perturbation theory made the derivation of this KdV-type equation possible. The motion of a wave, which starts as a KdV soliton, is studied according to the new equation in several cases by numerical simulations. The quantitative changes of a soliton's velocity and amplitude appear to be directly related to bottom variations. Changes of the soliton's velocity appear to be almost linearly anticorrelated with changes of water depth whereas correlation of variation of soliton's amplitude with changes of water depth looks less linear. When the bottom is flat, the new terms narrow down the family of exact solutions, but at least one single soliton survives. This is also checked by numerics.

nlin.PS↗
arXiv2014

New nonlinear equation of KdV-type for a non-flat bottom

In the paper a new nonlinear equation describing shallow water waves with the topography of the bottom directly taken into account is derived. This equation is valid in the weakly nonlinear, dispersive and long wavelength limit. Some examples of soliton motion for various bottom shapes obtained in numerical simulations according to the derived equation are presented.

nlin.PS↗
arXiv2012

A series approach to stochastic Volterra equations of convolution time

In the paper stochastic Volterra equations with noise terms driven by series of independent scalar Wiener processes are considered. In our study we use the resolvent approach to the equations under consideration. We give sufficient condition for the existence of strong solution to the class of stochastic Volterra equations of convolution type. We provide regularity of stochastic convolution, as well.

math.PR↗
arXiv2012

Temporal and spatial regularity of solutions to stochastic Volterra equations of convolution type

In the paper regularity of solutions to stochastic Volterra equations in a separable Hilbert space is studied. Sufficient conditions for the temporal and spatial regularity of stochastic convolutions corresponding to the equations under consideration are provided. The results obtained generalize some well-known regularity results for solutions to stochastic differential equations. The paper is a continuation of previous author's papers concerning stochastic Volterra equations.

math.PR↗
arXiv2007

Convolution type stochastic Volterra equations

The aim of this work is to present, in self-contained form, results concerning fundamental and the most important questions related to linear stochastic Volterra equations of convolution type. The paper is devoted to study the existence and some kind of regularity of solutions to stochastic Volterra equations in Hilbert space and the space of tempered distributions, as well. In recent years the theory of Volterra equations, particularly fractional ones, has undergone a big development. This is an emerging area of research with interesting mathematical questions and various important applications. The increasing interest in these equations comes from their applications to problems from physics and engeenering, particularly from viscoelasticity, heat conduction in materials with memory or electrodynamics with memory.

math.PR↗
arXiv2007

Strong solutions to stochastic Volterra equations

In this paper stochastic Volterra equations admitting exponentially bounded resolvents are studied. After obtaining convergence of resolvents, some properties for stochastic convolutions are studied. Our main result provide sufficient conditions for strong solutions to stochastic Volterra equations.

math.PR↗
arXiv2006

Convolution-type stochastic Volterra equations with additive fractional Brownian motion in Hilbert space

We consider convolution-type stochastic Volterra equations with additive Hilbert-valued fractional Brownian motion, $0<H<1$. We find the weak solution to this stochastic Volterra equation, and study its stochastic integral part, the stochastic convolution, which we show to be mean-zero Gaussian. We develop an Itô isometry for stochastic integrals with respect to a Hilbert-valued fractional Brownian motion, and use it to compute the covariance of the stochastic convolution. This formula, which uses fractional integrals and derivatives, generalizes the well-known formula from the case $H=1/2$.

math.PR↗
arXiv2006

Properties of convolutions arising in stochastic Volterra equations

The aim of this note is to provide some results for stochastic convolutions corresponding to stochastic Volterra equations in separable Hilbert space. We study convolution of the form $W^Ψ(t):=\int_0^t S(t-τ)Ψ(τ)dW(τ)$, $t\geq 0$, where $S(t), t\geq 0$, is so-called {\em resolvent} for Volterra equation considered,$Ψ$ is an appropriate process and $W$ is a cylindrical Wiener process.

math.PR↗
arXiv2006

On stochastic fractional Volterra equations in Hilbert space

In this paper stochastic Volterra equations admitting exponentially bounded resolvents are studied. After obtaining convergence of resolvents, some properties of stochastic convolutions are given. The paper provides a sufficient condition for a stochastic convolution to be a strong solution to a stochastic Volterra equation.

math.PR↗
arXiv2005

Stochastic Volterra equations of nonscalar type in Hilbert space

In the paper stochastic Volterra equations of nonscalar type in Hilbert space are studied. The aim of the paper is to provide some results on stochastic convolution and mild solutions to those Volterra equations. The motivation of the paper comes from a model of aging viscoelastic materials. The pseudo-resolvent approach is used.

math.PR↗
arXiv2005

Regularity of solutions to stochastic Volterra equations with infinite delay

In this article we give necessary and sufficient conditions providing regularity of solutions to stochastic Volterra equations with infinite delay on a $d$-dimensional torus. The harmonic analysis techniques and stochastic integration in function spaces are used. The work applies to both the stochastic heat and wave equations.

math.PR↗
arXiv2005

Maximal type inequalities for linear stochastic Volterra equations

The note is devoted to estimates for convolutions appearing in some class of stochastic Volterra equations. Two maximal inequalities and exponential tail estimate are proved by the fractional method of infinite dimensional stochastic calculus. The paper extends on non-semigroup case some results obtained earlier for semigroups.

math.PR↗