arXiv subjects

G. A. Tsolias

Publications and source records attributed to G. A. Tsolias.

8 recordsLinked to original sources

arXiv2024

Fractional Solitons: A Homotopic Continuation from the Biharmonic to the Harmonic $ϕ^4$ Model

In the present work we explore the path from a harmonic to a biharmonic PDE of Klein-Gordon type from a continuation/bifurcation perspective. More specifically, we make use of the Riesz fractional derivative as a tool that allows us to interpolate between these two limits. We illustrate, in particular, how the coherent kink structures existing in these models transition from the exponential tail of the harmonic operator case, via the power-law tails of intermediate fractional orders, to the oscillatory exponential tails of the biharmonic model. Importantly, we do not limit our considerations to the single kink case, but extend to the kink-antikink pair, finding an intriguing cascade of saddle-center bifurcations happening exponentially close to the biharmonic limit. Our analysis clearly explains the transition between the infinitely many stationary soliton pairs of the biharmonic case and the absence of even a single such pair in the harmonic limit. The stability of the different configurations obtained and the associated dynamics and phase portraits are also analyzed.

nlin.PS↗

arXiv2022

Kink-Antikink Interaction Forces and Bound States in a nonlinear Schr{ö}dinger Model with Quadratic and Quartic dispersion

In the present work we explore the competition of quadratic and quartic dispersion in producing kink-like solitary waves in a model of the nonlinear Schr{ö}dinger type bearing cubic nonlinearity. We present the first 6 families of multikink solutions and explore their bifurcations as the strength of the quadratic dispersion is varied. We reveal a rich bifurcation structure for the system, connecting two-kink states with states involving 4-, as well as 6-kinks. The stability of all of these states is explored. For each family, we discuss a ``lower branch'' adhering to the energy landscape of the 2-kink states. We also, however, study in detail the ``upper branches'' bearing higher numbers of kinks. In addition to computing the stationary states and analyzing their stability within the partial differential equation model, we develop an effective particle ordinary differential equation theory that is shown to be surprisingly efficient in capturing the kink equilibria and normal (as well as unstable) modes. Finally, the results of the bifurcation analysis are corroborated by means of direct numerical simulations involving the excitation of the states in a targeted way in order to explore their instability-induced dynamics.

nlin.PS↗

arXiv2022

Mixed dispersion nonlinear Schrödinger equation in higher dimensions: theoretical analysis and numerical computations

In the present work we provide a characterization of the ground states of a higher-dimensional quadratic-quartic model of the nonlinear Schr{ö}dinger class with a combination of a focusing biharmonic operator with either an isotropic or an anisotropic defocusing Laplacian operator (at the linear level) and power-law nonlinearity. Examining principally the prototypical example of dimension $d=2$, we find that instability arises beyond a certain threshold coefficient of the Laplacian between the cubic and quintic cases, while all solutions are stable for powers below the cubic. Above the quintic, and up to a critical nonlinearity exponent $p$, there exists a progressively narrowing range of stable frequencies. Finally, above the critical $p$ all solutions are unstable. The picture is rather similar in the anisotropic case, with the difference that even before the cubic case, the numerical computations suggest an interval of unstable frequencies. Our analysis generalizes the relevant observations for arbitrary combinations of Laplacian prefactor $b$ and nonlinearity power $p$.

nlin.PS↗

arXiv2021

Dark solitons under higher order dispersion

We show theoretically that dark solitons can exist in the presence of pure quartic dispersion, and also in the presence of both quadratic and quartic dispersive effects, displaying a much greater variety of possible solutions and dynamics than for pure quadratic dispersion. The interplay of the two dispersion orders may lead to oscillatory non-vanishing tails, which enables the possibility of bound, potentially stable, multi-soliton states. Dark soliton-like states which connect to low amplitude oscillations are also shown to be possible. Dynamical evolution results corroborate the stability picture obtained, and possible avenues for dark soliton generation are explored.

nlin.PS↗

arXiv2020

Kink-Antikink Interaction Forces and Bound States in a $ϕ^4$ Model with Quadratic and Quartic dispersion

We consider the interaction of solitary waves in a model involving the well-known $ϕ^4$ Klein-Gordon theory, but now bearing both Laplacian and biharmonic terms with different prefactors. As a result of the competition of the respective linear operators, we obtain three distinct cases as we vary the model parameters. In the first the biharmonic effect dominates, yielding an oscillatory inter-wave interaction; in the third the harmonic effect prevails yielding exponential interactions, while we find an intriguing linearly modulated exponential effect in the critical second case, separating the above two regimes. For each case, we calculate the force between the kink and antikink when initially separated with sufficient distance. Being able to write the acceleration as a function of the separation distance, and its corresponding ordinary differential equation, we test the corresponding predictions, finding very good agreement, where appropriate, with the corresponding partial differential equation results. Where the two findings differ, we explain the source of disparities. Finally, we offer a first glimpse of the interplay of harmonic and biharmonic effects on the results of kink-antikink collisions and the corresponding single- and multi-bounce windows.

nlin.PS↗

arXiv2008

Classical evolution of fractal measures generated by a scalar field on the lattice

We investigate the classical evolution of a $ϕ^4$ scalar field theory, using in the initial state random field configurations possessing a fractal measure expressed by a non-integer mass dimension. These configurations resemble the equilibrium state of a critical scalar condensate. The measures of the initial fractal behavior vary in time following the mean field motion. We show that the remnants of the original fractal geometry survive and leave an imprint in the system time averaged observables, even for large times compared to the approximate oscillation period of the mean field, determined by the model parameters. This behavior becomes more transparent in the evolution of a deterministic Cantor-like scalar field configuration. We extend our study to the case of two interacting scalar fields, and we find qualitatively similar results. Therefore, our analysis indicates that the geometrical properties of a critical system initially at equilibrium could sustain for several periods of the field oscillations in the phase of non-equilibrium evolution.

physics.comp-ph↗

arXiv2007

Classical evolution of fractal measures on the lattice

We consider the classical evolution of a lattice of non-linear coupled oscillators for a special case of initial conditions resembling the equilibrium state of a macroscopic thermal system at the critical point. The displacements of the oscillators define initially a fractal measure on the lattice associated with the scaling properties of the order parameter fluctuations in the corresponding critical system. Assuming a sudden symmetry breaking (quench), leading to a change in the equilibrium position of each oscillator, we investigate in some detail the deformation of the initial fractal geometry as time evolves. In particular we show that traces of the critical fractal measure can sustain for large times and we extract the properties of the chain which determine the associated time-scales. Our analysis applies generally to critical systems for which, after a slow developing phase where equilibrium conditions are justified, a rapid evolution, induced by a sudden symmetry breaking, emerges in time scales much shorter than the corresponding relaxation or observation time. In particular, it can be used in the fireball evolution in a heavy-ion collision experiment, where the QCD critical point emerges, or in the study of evolving fractals of astrophysical and cosmological scales, and may lead to determination of the initial critical properties of the Universe through observations in the symmetry broken phase.

physics.comp-ph↗

arXiv2006

An efficient algorithm simulating a macroscopic system at the critical point

It is well known that conventional simulation algorithms are inefficient for the statistical description of macroscopic systems exactly at the critical point due to the divergence of the corresponding relaxation time (critical slowing down). On the other hand the dynamics in the order parameter space is simplified significantly in this case due to the onset of self-similarity in the associated fluctuation patterns. As a consequence the effective action at the critical point obtains a very simple form. In the present work we show that this simplified action can be used in order to simulate efficiently the statistical properties of a macroscopic system exactly at the critical point. Using the proposed algorithm we generate an ensemble of configurations resembling the characteristic fractal geometry of the critical system related to the self-similar order parameter fluctuations. As an example we simulate the one-component real scalar field theory at the transition point $T=T_c$ as a representative system belonging to the $3-D$ Ising universality class.

physics.comp-ph↗