Computation and Stability of TravelingWaves in Second Order Evolution Equations

The topic of this paper are nonlinear traveling waves occuring in a system of damped waves equations in one space variable. We extend the freezing method from first to second order equations in time. When applied to a Cauchy problem, this method generates a comoving frame in which the solution becomes stationary. In addition it generates an algebraic variable which converges to the speed of the wave, provided the original wave satisfies certain spectral conditions and initial perturbations are sufficiently small. We develop a rigorous theory for this effect by recourse to some recent nonlinear stability results for waves in first order hyperbolic systems. Numerical computations illustrate the theory for examples of Nagumo and FitzHugh-Nagumo type

Freezing traveling and rotatingwaves in second order evolution equations

In this paper we investigate the implementation of the so-called freezing method for second order wave equations in one and several space dimensions. The method converts the given PDE into a partial differential algebraic equation which is then solved numerically. The reformulation aims at separating the motion of a solution into a co-moving frame and a profile which varies as little as possible. Numerical examples demonstrate the feasability of this approach for semilinear wave equations with sufficient damping. We treat the case of a traveling wave in one space dimension and of a rotating wave in two space dimensions. In addition, we investigate in arbitrary space dimensions the point spectrum and the essential spectrum of operators obtained by linearizing about the profile, and we indicate the consequences for the nonlinear stability of the wave

Putting ourselves in another’s skin: using the plasticity of self-perception to enhance empathy and decrease prejudice

The self is one the most important concepts in social cognition and plays a crucial role in determining questions such as which social groups we view ourselves as belonging to and how we relate to others. In the past decade, the self has also become an important topic within cognitive neuroscience with an explosion in the number of studies seeking to understand how different aspects of the self are represented within the brain. In this paper, we first outline the recent research on the neurocognitive basis of the self and highlight a key distinction between two forms of self-representation. The first is the “bodily” self, which is thought to be the basis of subjective experience and is grounded in the processing of sensorimotor signals. The second is the “conceptual” self, which develops through our interactions of other and is formed of a rich network of associative and semantic information. We then investigate how both the bodily and conceptual self are related to social cognition with an emphasis on how self-representations are involved in the processing and creation of prejudice. We then highlight new research demonstrating that the bodily and conceptual self are both malleable and that this malleability can be harnessed in order to achieve a reduction in social prejudice. In particular, we will outline strong evidence that modulating people’s perceptions of the bodily self can lead to changes in attitudes at the conceptual level. We will highlight a series of studies demonstrating that social attitudes towards various social out-groups (e.g. racial groups) can lead to a reduction in prejudice towards that group. Finally, we seek to place these findings in a broader social context by considering how innovations in virtual reality technology can allow experiences of taking on another’s identity are likely to become both more commonplace and more convincing in the future and the various opportunities and risks associated with using such technology to reduce prejudice

Spatial decay and spectral properties of rotating waves in parabolic systems

Otten D. Spatial decay and spectral properties of rotating waves in parabolic systems. Berichte aus der Mathematik. Aachen: Shaker; 2014

Spatial decay of rotating waves in reaction diffusion systems

Beyn W-J, Otten D. Spatial decay of rotating waves in reaction diffusion systems. Dynamics of Partial Differential Equations. 2016;13(3):191-240.In this paper we study nonlinear problems for Ornstein-Uhlenbeck operators A Delta v(x) + + f(v(x)) = 0, x is an element of R-d, d >= 2, where the matrix A is an element of R-N,R- N is diagonalizable and has eigenvalues with positive real part, the map f : R-N -> R-N is sufficiently smooth and the matrix S is an element of R-d,R- d in the unbounded drift term is skew-symmetric. Nonlinear problems of this form appear as stationary equations for rotating waves in time-dependent reaction diffusion systems. We prove under appropriate conditions that every bounded classical solution v(*) of the nonlinear problem, which falls below a certain threshold at infinity, already decays exponentially in space, in the sense that v(*) belongs to an exponentially weighted Sobolev space W-theta(1, p) (R-d, R-N). Several extensions of this basic result are presented: to complex-valued systems, to exponential decay in higher order Sobolev spaces and to pointwise estimates. We also prove that every bounded classical solution v of the eigen-value problem A Delta v(x) + + Df(v(*)(x))v(x) = lambda v(x), x is an element of R-d, d >= 2, decays exponentially in space, provided Re lambda lies to the right of the essential spectrum. As an application we analyze spinning soliton solutions which occur in the Ginzburg-Landau equation. Our results form the basis for investigating nonlinear stability of rotating waves in higher space dimensions and truncations to bounded domains

Spectral analysis of localized rotating waves in parabolic systems

Beyn W-J, Otten D. Spectral analysis of localized rotating waves in parabolic systems. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences. 2018;376(2117): 28.In this paper, we study the spectra and Fredholm properties of Ornstein-Uhlenbeck operators Lv(x): = A Delta v(x) + + Df(v(x)(x))v(x), x is an element of R-d , d >= 2, where v : R-d -> R-m is the profile of a rotating wave satisfying v (x) -> v(infinity) is an element of R-m as vertical bar x vertical bar -> infinity, the map f : R-m -> R-m is smooth and the matrix A is an element of R-m,R-m eigenvalues with positive real parts and commutes with the limit matrix Df (v(infinity)). The matrix S. Rd, d is assumed to be skew- symmetric with eigenvalues (lambda(1),...,lambda(d)) = (+/- i sigma(1),...,+/- i sigma(k),0,..., 0). The spectra of these linearized operators are crucial for the nonlinear stability of rotating waves in reaction-diffusion systems. We prove under appropriate conditions that every lambda is an element of C satisfying the dispersion relation det(lambda I-m + eta(2)A - Df(v(infinity)) + i I-m) = 0 for soe=me eta is an element of R and n is an element of Z(k), sigma = (sigma(1),...,sigma(k)) is an element of R-k belongs to the essential spectrum sigma(ess)(L) in L-p. For values Re lambda to the right of the spectral bound for Df (v(infinity)), we show that the operator lambda I - L is Fredholm of index 0, solve the identification problem for the adjoint operator (lambda I - L)* and formulate the Fredholm alternative. Moreover, we show that the set sigma(S) boolean OR {lambda(i) + lambda(i) : lambda(i), lambda(j is an element of) sigma(S), 1 <= i <= j <= d} belongs to the point spectrum sigma(pt)(L) in L-p. We determine the associated eigenfunctions and show that they decay exponentially in space. As an application, we analyse spinning soliton solutions which occur in the Ginzburg-Landau equation and compute their numerical spectra as well as associated eigenfunctions. Our results form the basis for investigating the nonlinear stability of rotating waves in higher space dimensions and truncations to bounded domains. This article is part of the themed issue 'tability of nonlinear waves and patterns and related topics'

Computation and Stability of Traveling Waves in Second Order Evolution Equations

Beyn W-J, Otten D, Rottmann-Matthes J. Computation and Stability of Traveling Waves in Second Order Evolution Equations. SIAM Journal on Numerical Analysis. 2018;56(3):1786-1817.The topic of this paper is nonlinear traveling waves occuring in a system of damped wave equations in one space variable. We extend the freezing method from first to second order equations in time. When applied to a Cauchy problem, this method generates a co-moving frame in which the solution becomes stationary. In addition, it generates an algebraic variable which converges to the speed of the wave, provided the original wave satisfies certain spectral conditions and initial perturbations are sufficiently small. We develop a rigorous theory for this effect by recourse to some recent nonlinear stability results for waves in first order hyperbolic systems. Numerical computations illustrate the theory for examples of Nagumo and FitzHugh-Nagumo type

Supplementary Information from Spectral analysis of localized rotating waves in parabolic systems

In this paper, we study the spectra and Fredholm properties of Ornstein–Uhlenbeck operator