One-dimensional Evolution Equation Research Articles

Short-wavelength soliton in a fully degenerate quantum plasma

We present a novel one-dimensional nonlinear evolution equation governing the dynamics short-wavelength longitudinal waves in a nonrelativistic fully degenerate quantum plasma using kinetic equation for the Wigner function. The linear dispersion of the equation has a form of “zero sound” ∼k exp (−k2), where k is the wave number, and it strongly differs from previously known nonlinear evolution equations. We numerically find the corresponding soliton solutions and demonstrate that the collisions between three solitons turn out to be elastic, resulting only in phase shifts.

A 1D kinetic model for cosmic microwave background comptonization

This work presents a novel derivation of the expressions that describe the distortions of the CMB curve due to the interactions between photons and the electrons present in dilute ionized systems. In this approach, a simplified a one-dimensional evolution equation for the photon number occupation is applied in order to describe the aforemationed interaction. This methodology allows to emphasize the physical features ot the Sunyaev-Zeldovich effect and suggests the existence of links between basic statistical physics and complex applications involving radiative processes.

Analysis of a perturbed Cahn–Hilliard model for Langmuir–Blodgett films

An advective Cahn–Hilliard model motivated by thin film formation is studied in this paper. The one-dimensional evolution equation under consideration includes a transport term, whose presence prevents from identifying a gradient flow structure. Existence and uniqueness of solutions, together with continuous dependence on the initial data and an energy equality are proved by combining a minimizing movement scheme with a fixed point argument. Finally, it is shown that, when the contribution of the transport term is small, the equation possesses a global attractor and converges, as the transport term tends to zero, to a purely diffusive Cahn–Hilliard equation.

On the $ L^p $ regularity of solutions to the generalized Hunter-Saxton system

The generalized Hunter-Saxton system comprises several well-kno-wn models from fluid dynamics and serves as a tool for the study of fluid convection and stretching in one-dimensional evolution equations. In this work, we examine the global regularity of periodic smooth solutions of this system in \begin{document}$ L^p $\end{document} , \begin{document}$ p \in [1,\infty) $\end{document} , spaces for nonzero real parameters \begin{document}$ (\lambda,\kappa) $\end{document} . Our results significantly improve and extend those by Wunsch et al. [ 29 , 30 , 31 ] and Sarria [ 23 ]. Furthermore, we study the effects that different boundary conditions have on the global regularity of solutions by replacing periodicity with a homogeneous three-point boundary condition and establish finite-time blowup of a local-in-time solution of the resulting system for particular values of the parameters.

Asymptotic Solutions of One-Dimensional Linear Evolution Equations for Surface Waves with Account for Surface Tension

Asymptotic Solutions of One-Dimensional Linear Evolution Equations for Surface Waves with Account for Surface Tension

Asymptotic Solutions of One-Dimensional Linear Evolution Equations for Surface Waves with Account for Surface Tension

Asymptotic Solutions of One-Dimensional Linear Evolution Equations for Surface Waves with Account for Surface Tension

Self-similar rupture of thin films of power-law fluids on a substrate

Thinning and rupture of a thin film of a power-law fluid on a solid substrate under the balance between destabilizing van der Waals pressure and stabilizing capillary pressure is analysed. In a power-law fluid, viscosity is not constant but is proportional to the deformation rate raised to the $n-1$ power, where $0<n\leqslant 1$ is the power-law exponent ($n=1$ for a Newtonian fluid). In the first part of the paper, use is made of the slenderness of the film and the lubrication approximation is applied to the equations of motion to derive a spatially one-dimensional nonlinear evolution equation for film thickness. The variation with time remaining until rupture of the film thickness, the lateral length scale, fluid velocity and viscosity is determined analytically and confirmed by numerical simulations for both line rupture and point rupture. The self-similarity of the numerically computed film profiles in the vicinity of the location where the film thickness is a minimum is demonstrated by rescaling of the transient profiles with the scales deduced from theory. It is then shown that, in contrast to films of Newtonian fluids undergoing rupture for which inertia is always negligible, inertia can become important during thinning of films of power-law fluids in certain situations. The critical conditions for which inertia becomes important and the lubrication approximation is no longer valid are determined analytically. In the second part of the paper, thinning and rupture of thin films of power-law fluids in situations when inertia is important are simulated by solving numerically the spatially two-dimensional, transient Cauchy momentum and continuity equations. It is shown that as such films continue to thin, a change of scaling occurs from a regime in which van der Waals, capillary and viscous forces are important to one where the dominant balance of forces is between van der Waals, capillary and inertial forces while viscous force is negligible.

Multidomain Legendre–Galerkin Chebyshev-collocation method for one-dimensional evolution equations with discontinuity

The multidomain Legendre–Galerkin Chebyshev-collocation method is considered to solve one-dimensional linear evolution equations with two nonhomogeneous jump conditions. The scheme treats the first jump condition essentially and the second one naturally. We adopt appropriate base functions to deal with interfaces. The proposed method can be implemented in parallel. Error analysis shows that the approach has an optimal convergence rate. The proposed method is also applied to computing the one-dimensional Maxwell equation and the one-dimensional two phase Stefan problem, respectively. Numerical examples are given to confirm the theoretical analysis.

Self-similarity and scaling transitions during rupture of thin free films of Newtonian fluids

Rupture of thin liquid films is crucial in many industrial applications and nature such as foam stability in oil-gas separation units, coating flows, polymer processing, and tear films in the eye. In some of these situations, a liquid film may have two free surfaces (referred to here as a free film or a sheet) as opposed to a film deposited on a solid substrate that has one free surface. The rupture of such a free film or a sheet of a Newtonian fluid is analyzed under the competing influences of inertia, viscous stress, van der Waals pressure, and capillary pressure by solving a system of spatially one-dimensional evolution equations for film thickness and lateral velocity. The dynamics close to the space-time singularity where the film ruptures is asymptotically self-similar and, therefore, the problem is also analyzed by reducing the transient partial differential evolution equations to a corresponding set of ordinary differential equations in similarity space. For sheets with negligible inertia, it is shown that the dominant balance of forces involves solely viscous and van der Waals forces, with capillary force remaining negligible throughout the thinning process in a viscous regime. On the other hand, for a sheet of an inviscid fluid for which the effect of viscosity is negligible, it is shown that the dominant balance of forces is between inertial, capillary, and van der Waals forces as the film evolves towards rupture in an inertial regime. Real fluids, however, have finite viscosity. Hence, for real fluids, it is further shown that the viscous and the inertial regimes are only transitory and can only describe the initial thinning dynamics of highly viscous and slightly viscous sheets, respectively. Moreover, regardless of the fluid’s viscosity, it is shown that for sheets that initially thin in either of these two regimes, their dynamics transition to a late stage or final inertial-viscous regime in which inertial, viscous, and van der Waals forces balance each other while capillary force remains negligible, in accordance with the results of Vaynblat, Lister, and Witelski.

L2-stability of traveling wave solutions to nonlocal evolution equations

Stability of the traveling wave solution to a general class of one-dimensional nonlocal evolution equations is studied in L2-spaces, thereby providing an alternative approach to the usual spectral analysis with respect to the supremum norm. We prove that the linearization around the traveling wave solution satisfies a Lyapunov-type stability condition in a weighted space L2(ρ) for a naturally associated density ρ. The result can be applied to obtain stability of the traveling wave solution under stochastic perturbations of additive or multiplicative type. For small wave speeds, we also prove an alternative Lyapunov-type stability condition in L2(m), where m is the symmetrizing density for the traveling wave operator, which allows to derive a long-term stochastic stability result.

On general form of the Tanh method and its application to nonlinear partial differential equations

The tanh method is used to compute travelling waves solutions of one-dimensional nonlinear wave and evolution equations. The technique is based on seeking travelling wave solutions in the form of a finite series in tanh. In this article, we introduce a new general form of tanh transformation and solve well-known nonlinear partial differential equations in which tanh method becomes weaker in the sense of obtaining general form of solutions.

Analysis of image processing model based on pixels of solitary waves

Through the study on nonlinear effect of digital image pixel grid,the one-dimensional and twodimensional time evolution equations model of nonlinear effect between pixels were established,the analysis of the equation model shows that the image was discrete in space,but on the structure of mutual information was a continuous differential difference equations,and the equation had analytic solitary wave solutions. The model was used to study the interaction of image pixels between two solitary waves,pixel solitary wave of two solitary wave solutions were given,and used the interaction of solitary waves of the pixel to study the performance of model,it was found that pixel solitary wave can still maintain their properties unchanged after the interaction,so pixel itself can be replaced by pixel solitary wave. To study the interaction situation of pixel solitary wave,it was found that pixel solitary wave amplitude was nonlinear superposition of the two,so it could be used as the result of the influence. The phase change of pixel solitary wave can map for direction and provide direction information of the image. The experiment result shows that the model can be used in image filtering,although the smoothness is not optimal,image details retain more.

Electrostatic Effects on Linear and Nonlinear Waves in Hanging Film Flows

The dynamics of a liquid film wetting the underside of a semi-infinite horizontal substrate are investigated. Gravity causes the film to be unstable and the objective of this work is to control such instabilities by using horizontal electric fields. The film wets a semi-infinite dielectric solid block and is bounded below by air. The electric fields from these two regions are coupled to those in the liquid film and contribute to the hydrodynamics through the interfacial boundary conditions by introducing electric Maxwell stresses in the stress tensor. As a result novel nonlocal terms arise and their effect is studied both in the linear regime as well as nonlinearly through a derivation of a system of long-wave one-dimensional evolution equations describing the interfacial position and the leading order horizontal velocity in the film. When surface tension is present there exists a band of unstable wavenumbers rendering the film long-wave unstable. It is shown that the size of this band decreases asymptotically to zero as the electric field strength is increased (physically this means that any system of finite horizontal length can be fully stabilized by a sufficiently strong electric field). The derived nonlinear system supports nonlinear coherent structures in the form of traveling waves of permanent form, and such solutions are computed numerically for a range of parameters. The production of such traveling waves may be useful in enhancing the transport properties in hanging film systems.

Nonlinear Wave Packets in Deformed Honeycomb Lattices

The spectrum of a Schrödinger operator with a perfect honeycomb lattice potential has special points, called Dirac points, where the lowest two branches of the spectrum touch. Deformations can result in the merging and disappearance of the Dirac points, and the originally intersecting dispersion relation branches separate. Corresponding to these deformations, nonlinear envelope equations are derived and their dynamics are studied. In the region where Dirac points exist, a maximally balanced equation is derived which has limits to a nonlinear Schrödinger--Kadomtsev--Petviashvili (NLSKP)-type equation and its dispersionless reduction. When the Dirac points disappear and a gap opens, a different maximally balanced equation is derived which has the NLSKP equation and a one-dimensional nonlocal evolution equation as limits. When the gap is sufficiently wide, a nonlinear Dirac equation with nonzero mass and a nonlinear Schrödinger focusing-defocusing system are found. The latter two equations admit nonlinear localized modes. Typical dynamical behaviors of the effective envelope equations are presented.

Linear stability of finite-amplitude capillary waves on water of infinite depth

AbstractWe study the linear stability of the exact deep-water capillary wave solution of Crapper (J. Fluid Mech., vol. 2, 1957, pp. 532–540) subject to two-dimensional perturbations (both subharmonic and superharmonic). By linearizing a set of exact one-dimensional non-local evolution equations, a stability analysis is performed with the aid of Floquet theory. To validate our results, the exact evolution equations are integrated numerically in time and the numerical solutions are compared with the time evolution of linear normal modes. For superharmonic perturbations, contrary to Hogan (J. Fluid Mech., vol. 190, 1988, pp. 165–177), who detected two bubbles of instability for intermediate amplitudes, our results indicate that Crapper’s capillary waves are linearly stable to superharmonic disturbances for all wave amplitudes. For subharmonic perturbations, it is found that Crapper’s capillary waves are unstable, and our results generalize to the highly nonlinear regime the analysis for small amplitudes presented by Chen & Saffman (Stud. Appl. Maths, vol. 72, 1985, pp. 125–147).

Calculation of poloidal velocity in the tokamak plasma with allowance for density inhomogeneity and diamagnetic drift of ions

A one-dimensional evolution equation for the angle-averaged poloidal momentum of the tokamak plasma is derived in the framework of reduced magnetohydrodynamics with allowance for density inhomogeneity and diamagnetic drift of ions. In addition to fluctuations of the E × B drift velocity, the resulting turbulent Reynolds stress tensor includes fluctuations of the ion density and ion pressure, as well as turbulent radial fluxes of particles and heat. It is demonstrated numerically by using a particular example that the poloidal velocity calculated using the refined one-dimensional evolution equation differs substantially from that provided by the simplified model. When passing to the new model, both the turbulent Reynolds force and the Stringer-Winsor force increase, which leads to an increase in the amplitude of the ion poloidal velocity. This, in turn, leads to a decrease in turbulent fluxes of particles and heat due to the effect of shear decorrelation.

Soliton ensembles and solitonic structures

The article deals with the analysis of three complicated one-dimensional evolution equations, the solutions of which can be classified as soliton ensembles or solitonic structures. The case studies from the physical viewpoint are: (i) martensitic–austenitic alloys; (ii) hyperelastic rod and (iii) granular materials. The corresponding evolution equations governing the propagation of longitudinal waves include higher order nonlinear and dispersive terms and are nonintegrable. Numerical simulation is carried out by the pseudospectral method. The first class of solutions involves solitons with nonvanishing oscillatory tails and wave packets called solitonic structures. The second class includes special soliton ensembles or more exactly – plaited solitons. The emergence of such entities and their interaction demonstrate the solitonic character of waves.

Effective one-dimensional description of confined diffusion biased by a transverse gravitational force

Diffusion of pointlike noninteracting particles in a two-dimensional channel of varying cross section is considered. The particles are biased by a constant force in the transverse direction. A recurrence mapping procedure is applied, which enables the derivation of an effective one-dimensional (1D) evolution equation that governs the 1D density of the particles in the channel. In the limit of stationary flow, an extended Fick-Jacobs equation is reached, which is corrected by an effective diffusion coefficient D(x) that depends on the longitudinal coordinate x. The result is an approximate formula for D(x) that also involves the influence of the transverse force. The calculations are verified by the stationary diffusion in a linear cone, which is exactly solvable.

Mapping of diffusion in a channel with soft walls

We study diffusion of pointlike particles biased toward the x axis by a quadratic potential U(x,y)=κ(x)y². This system mimics a channel with soft walls of some varying (effective) cross section A(x), depending on the stiffness κ(x). We show that diffusion in this geometry can also be mapped rigorously onto the longitudinal coordinate x by a procedure known for channels with hard walls [P. Kalinay and J. K. Percus, Phys. Rev. E 74, 041203 (2006)]; i.e., we arrive at a one-dimensional evolution equation of the Fick-Jacobs type. On the other hand, the calculation presented serves as a prototype for mapping of the Smoluchowski equation with a wide class of potentials U(x,y) varying in both the longitudinal as well as the transverse directions, which is necessary for understanding, e.g., stochastic resonance.

Asymptotics for a free boundary model in price formation

We study the asymptotics for a large time of solutions to a one-dimensional parabolic evolution equation with non-standard measure-valued right hand side, that involves derivatives of the solution computed at a free boundary point. The problem is a particular case of a mean-field free boundary model proposed by Lasry–Lions on price formation and dynamic equilibria. The main step in the proof is based on the fact that the free boundary disappears in the linearized problem, thus it can be treated as a perturbation through semigroup theory. This requires a delicate choice for the function spaces since higher regularity is needed near the free boundary. We show global existence for solutions with initial data in a small neighborhood of any equilibrium point, and exponential decay towards a stationary state. Moreover, the family of equilibria of the equation is stable, as follows from center manifold theory.