One-parameter Family Of Solutions Research Articles

Defects composed of kinks and Q -balls: Analytical solutions and stability

In this paper all the defect-type solutions in a family of scalar field theories with a real and a complex field in (1+1) dimensional Minkowski spacetime have been analytically identified. Three types of solutions have been found: (a) topological kinks without the presence of $Q$-balls, (b) defects which consist of a topological kink coupled with a $Q$-ball and (c) a one-parameter family of solutions where a $Q$-ball is combined with a non-topological soliton. The properties of these solutions and its linear stability are also discussed.

Inhibitor-Induced Wavetrains and Spiral Waves in an Extended FitzHugh-Nagumo Model of Nerve Cell Dynamics.

It is well known that the FitzHugh-Nagumo model is one of the simplified forms of the four-variable Hodgkin-Huxley model that can reflect most of the significant phenomena of nerve cell action potential. However, this model cannot capture the irregular action potentials of sufficiently large periods in a one-parameter family of solutions. Motivated by this, we propose a modified FitzHugh-Nagumo reaction-diffusion system by changing its recovery kinetics. First, we investigate the parameter regime to know the existence of the wavetrains. Second, we conceive the occurrence of Eckhaus bifurcations of solutions that divide the solution region into two parts. The essential spectra at different grid points explore the occurrence of bifurcations of the waves. We find that the wavetrains of sufficiently large periods cross the stability boundary. This characteristic phenomenon is absent in the standard FitzHugh-Nagumo model. Finally, we observe a reasonable agreement between the direct PDE simulations and the solutions in the traveling wave ODEs. Furthermore, the model exhibits spiral wave for monotone and non-monotone cases that agrees with the waves observed in cellular activity.

Scale-dependent Rayleigh–Taylor dynamics with variable acceleration in a finite-sized domain for three-dimensional flows

Three-dimensional Rayleigh–Taylor instability (RTI) with the time-varying acceleration in a finite domain is investigated in a systematic framework. The acceleration magnitude follows a power law in time with an exponent greater than −2. Applying the group theory, the instabilities are demonstrated considering the irreducible representations for observable periodic structures with a square symmetry in the plane normal to the acceleration. We derive the dynamical system and illustrate the universal form of the solutions in the linear and nonlinear regimes. The scale-dependent dynamics are shown to be single scale and multiscale in the two regimes, respectively. For the nonlinear regime solutions, fundamental scales are derived bridging the solutions in the finite- and infinite-sized domains. Special solutions for bubbles and spikes are identified from a one-parameter family of solutions. The effect of domain confinement is that the velocity and curvature decreases and shear increases as the domain size is reduced. The theory provides predictions for the flow field and demonstrates the interfacial behavior of RTI. Our results are in good agreement with the prior studies and also provide new benchmarks for experiments and simulations.

Stability analysis of periodic traveling waves in a model of vegetation patterns in semi-arid ecosystems

Banded vegetation pattern is a striking feature of self-organized ecosystems. In particular, vegetation patterns in semi-arid ecosystems are typical on hillsides, orienting parallel to the contours. The present study concerns a system of coupled reaction–advection–diffusion equations model for this phenomenon and studies the existence and stability of periodic traveling waves in a one-parameter family of solutions. Specifically, we consider a parameter region in which vegetation patterns occur and subdivide into stable and unstable regions as solutions of the model equations. Our numerical results show that the periodic traveling wave changes their stability by Eckhaus (sideband) bifurcation. We discuss the variations of the wavelength, wave speed as well as the conditions of the rainfall parameter by using linear analysis. We also explore how the solution patterns grow when the bifurcation parameter is changing slowly. In order to compare this result with the spatiotemporal pattern in the direct simulation, we show that when it passes a critical value of rainfall parameter a stable pattern becomes unstable and finally disappear. In addition, we investigate the existence and stability of periodic traveling waves as a function of water transport parameter.

Nonlinear and nonlocal eigenvalue problems: variational existence, decay properties, approximation, and universal scaling limits

We study a class of nonlinear eigenvalue problems which involves a convolution operator as well as a superlinear nonlinearity. Our variational existence proof is based on constrained optimization and provides a one-parameter family of solutions with positive eigenvalues and unimodal eigenfunctions. We also discuss the decay properties and the numerical computations of those eigenfunctions, and conclude with two asymptotic results concerning small and large eigenvalues.

Hybrid Metric-Palatini Gravity: Black Holes, Wormholes, Singularities, and Instabilities

The hybrid metric-Palatini theory of gravity (HMPG), proposed in 2012 by T. Harko et al., is known to successfully describe both local (solar-system) and cosmological observations. We discuss static, spherically symmetric vacuum solutions of HMPG with the aid of its scalar-tensor representation. This scalar-tensor theory coincides with general relativity with a conformally coupled scalar field (which can be canonical or phantom), therefore the known solutions of this theory are re-interpreted in terms of HMPG. In particular, in the case of zero scalar field potential $V (\phi)$, such that both Riemannian and Palatini Ricci scalars are zero, generic asymptotically flat solutions either contain naked singularities or describe traversable wormholes, and there are only special cases of black hole solutions with extremal horizons. There is also a one-parameter family of solutions with an infinite number of extremal horizons between static regions. Examples of analytical solutions with nonzero potentials $V (\phi)$ are also described, among them black hole solutions with simple horizons which are generic but, for canonical scalars, they require (at least partly) negative potentials. With phantom scalars there are ``black universe'' solutions that lead beyond the horizon to an expanding universe instead of a singularity. Most of the solutions under consideration turn out to be unstable under scalar monopole perturbations, but some special black hole solutions are stable.

Generalised solutions to the Benjamin problem

We generalise the classical Benjamin solution (Benjamin, J. Fluid Mech. , vol. 31, 1968, pp. 209–248) modelling the flow in a horizontal duct of finite depth in situations where the flow contains a region spanning the depth of the duct, and a region in which the surface detaches from the ceiling of the duct as a free surface. It is shown that the Benjamin solution belongs to a one-parameter family of similar solutions, which are divided into two types: solutions that describe potential flows where the free surface of the fluid is deflected from the duct ceiling at a zero angle; and solutions that admit the formation of a vortex flow region in the vicinity of the point of fluid separation from the duct ceiling. It is shown that this one-parameter family of solutions is the limit of a two-parameter family of solutions in which part of the uniform flow energy is converted into energy of the small-scale fluid motion. Based on the local hydrostatic approximation, the applicability of the constructed solutions is discussed.

Spherically Symmetric Black Holes and Wormholes in Hybrid Metric-Palatini Gravity

The so-called hybrid metric-Palatini theory of gravity (HMPG), proposed in 2012 by T. Harko et al., is known to successfully describe both local (solar-system) and cosmological observations. This paper gives a complete description of static, spherically vacuum solutions of HMPG in the simplest case where its scalar-tensor representation has a zero scalar field potential $V(\phi)$, and both Riemannian ($R$) and Palatini ($\cal R$) Ricci scalars are zero. Such a scalar-tensor theory coincides with general relativity with a phantom conformally coupled scalar field as a source of gravity. Generic asymptotically flat solutions either contain naked central singularities or describe traversable wormholes, and there is a special two-parameter family of globally regular black hole solutions with extremal horizons. In addition, there is a one-parameter family of solutions with an infinite number of extremal horizons between static regions and a spherical radius monotonically changing from region to region. It is argued that the obtained black hole and wormhole solutions are unstable under monopole perturbations. As a by-product, it is shown that a scalar-tensor theory with $V(\phi)=0$, in which there is at least one nontrivial ($\phi \ne \rm const$) vacuum solution with $R\equiv 0$, necessarily reduces to a theory with a conformal scalar field (the latter may be usual or phantom).

Existence of Periodic Traveling Waves in the Klausmeier Model of Desertification and its Modification: A Comparative Study

Self-organized and spatially periodic banded vegetation patterns have been observed in many semi-arid ecosystems. In order to understand the mechanism of these patterns, we consider a system of reaction-advection-diffusion equations in a two-variable model of desertification. This work deals with the investigation of the existence of periodic traveling waves in a one-parameter family of solutions. In addition, we investigate the existence of periodic traveling waves as a function of water transport parameter in the model.
 GANIT J. Bangladesh Math. Soc.Vol. 38 (2018) 27-46

Numerical Bifurcation Analysis to Study Periodic Traveling Wave Solutions in a Model of Young Mussel Beds

Self-bottomed mussel beds are dominant feature of ecosystem-scale self-organization. Regular spatial patterns of mussel beds in inter-tidal zone are typical, aligned perpendicular to the average incoming tidal flow. In this paper, we consider a two-variable partial differential equations model of young mussel beds. Our aim is to study the existence and stability of periodic traveling waves in a one-parameter family of solutions. We consider a parameter regime to show pattern existence in the model of young mussel beds. In addition, it is found that the periodic traveling waves changes their stability by two ways: Hopf type and Eckhaus type. We explain this stability by the calculation of essential spectra at different grid points in the two-dimensional parameter plane.
 GANIT J. Bangladesh Math. Soc.Vol. 38 (2018) 1-10

Electrovacuum geometries in five dimensions

The Chong-Cvetic-Lu-Pope 5D rotating charged black hole proves to belong to a set of solutions to Einstein-Maxwell-Chern-Simons (EMCS) equations that share the electromagnetic potential and the Chern-Simons coupling constant but differ in the Kretschmann invariant. This one-parametric family of solutions is found by proposing a properly deformed Plebanski-Demianski Ansatz for modeling the metric tensor. While no black-hole solutions for other values of the Chern-Simons coupling constant are found within this Ansatz, another type of non-static electrovacuum solutions to 5D EMCS equations are obtained, namely Kundt spacetimes sourced by a pure-radiation field.

Spiral breakup in a RD system of cardiac excitation due to front–back interaction

We consider a two-variable partial differential equations model of cardiac excitation and study spiral wave instability in a one-parameter family of solutions. We investigate numerically the existence of periodic traveling wave solution and show the front and the back interaction far away from the bifurcation point in one dimension. In two dimensions, we show the emergence of a stable spiral pattern before the bifurcation point. The most complex spatiotemporal pattern is called ventricular fibrillation when the breakup of one spiral wave makes another wave and the medium becomes chaotic. We show spiral wave instability and periodic traveling wave instability in the same computational settings. It is found that the pattern of the front–back interaction in two dimensions is similar with that of in the one dimension.

Stationary convection–diffusion equation in an infinite cylinder

We study the existence and uniqueness of a solution to a linear stationary convection–diffusion equation stated in an infinite cylinder, Neumann boundary condition being imposed on the boundary. We assume that the cylinder is a junction of two semi-infinite cylinders with two different periodic regimes. Depending on the direction of the effective convection in the two semi-infinite cylinders, we either get a unique solution, or one-parameter family of solutions, or even non-existence in the general case. In the latter case we provide necessary and sufficient conditions for the existence of a solution.

Time-Periodic Einstein–Klein–Gordon Bifurcations of Kerr

We construct one-parameter families of solutions to the Einstein–Klein–Gordon equations bifurcating off the Kerr solution such that the underlying family of spacetimes are each an asymptotically flat, stationary, axisymmetric, black hole spacetime, and such that the corresponding scalar fields are non-zero and time-periodic. An immediate corollary is that for these Klein–Gordon masses, the Kerr family is not asymptotically stable as a solution to the Einstein–Klein–Gordon equations.

Applying Bayesian inference to Galileon solutions of the muon problem

We derive corrections to atomic energy levels from disformal couplings in Galileon theories. Through Bayesian inference, we constrain the cutoff radii and Galileon scale via these corrections. To connect different atomic systems, we assume the various cutoff radii related by a one-parameter family of solutions. This introduces a new parameter $\alpha$ which is also constrained. In this model, we predict shifts to muonic helium of $\delta E_{He^3}=1.97^{+9.28}_{-1.87}$ meV and $\delta E_{He^4}=1.69^{+9.25}_{-1.61}$ meV as well as for true muonium, $\delta E_{TM}=0.06^{+0.46}_{-0.05}$ meV.

Heat flux of a granular gas with homogeneous temperature

A steady state of granular gas with homogeneous granular temperature, no mass flow, and nonzero heat flux is studied. The state is created by applying an external position-dependent force or by enclosing the grains inside a curved 2D silo. At a macroscopic level, the state is identified with one solution to the inelastic Navier–Stokes equations, due to the coupling between the heat flux induced by the density gradient and the external force. In contrast, at the mesoscopic level, by exactly solving a BGK model or the inelastic Boltzmann equation in an approximate way, a one-parametric family of solutions is found. Molecular dynamics simulations of the system in the quasi-elastic limit are in agreement with the theoretical results.

Regular black holes inf(R)gravity coupled to nonlinear electrodynamics

We obtain a class of regular black hole solutions in four-dimensional $f(R)$ gravity, $R$ being the curvature scalar, coupled to a nonlinear electromagnetic source. The metric formalism is used and static spherically symmetric spacetimes are assumed. The resulting $f(R)$ and nonlinear electrodynamics functions are characterized by a one-parameter family of solutions which are generalizations of known regular black holes in general relativity coupled to nonlinear electrodynamics. The related regular black holes of general relativity are recovered when the free parameter vanishes, in which case one has $f(R)\propto R$. We analyze the regularity of the solutions and also show that there are particular solutions that violate only the strong energy condition

Alternans and Spiral Breakup in an Excitable Reaction-Diffusion System: A Simulation Study.

The determination of the mechanisms of spiral breakup in excitable media is still an open problem for researchers. In the context of cardiac electrophysiological activities, spiral breakup exhibits complex spatiotemporal pattern known as ventricular fibrillation. The latter is the major cause of sudden cardiac deaths all over the world. In this paper, we numerically study the instability of periodic planar traveling wave solution in two dimensions. The emergence of stable spiral pattern is observed in the considered model. This pattern occurs when the heart is malfunctioning (i.e., ventricular tachycardia). We show that the spiral wave breakup is a consequence of the transverse instability of the planar traveling wave solutions. The alternans, that is, the oscillation of pulse widths, is observed in our simulation results. Moreover, we calculate the widths of spiral pulses numerically and observe that the stable spiral pattern bifurcates to an oscillatory wave pattern in a one-parameter family of solutions. The spiral breakup occurs far below the bifurcation when the maximum and the minimum excited states become more distinct, and hence the alternans becomes more pronounced.

Gravity dual of supersymmetric gauge theories on a squashed five-sphere.

We present the gravity dual of large N supersymmetric gauge theories on a squashed five-sphere. The one-parameter family of solutions is constructed in Euclidean Romans F(4) gauged supergravity in six dimensions, and uplifts to massive type IIA supergravity. By renormalizing the theory with appropriate counterterms we evaluate the renormalized on-shell action for the solutions. We also evaluate the large N limit of the gauge theory partition function, and find precise agreement.

Finding a cyclide given three contact conditions

Dupin cyclides form a 9-dimensional set of surfaces which are, from the viewpoint of differential geometry, the simplest after planes and spheres. We prove here that, given three oriented contact conditions, there is in general no Dupin cyclide satisfying them, but if the contact conditions belong to a codimension one subset, then there is a one-parameter family of solutions.