System Of Nonlinear Evolution Equations Research Articles

Study of Nonlinear Evolution of Spacetime Fluctuations in Quantum Gravity Inflation for Deriving Primordial Spectrum

We study the evolution of quantum fluctuations of gravity around an inflationary solution in renormalizable quantum gravity, in which the initial scalar-fluctuation dominance is shown by the background-free nature expressed by a special conformal invariance. Inflation ignites at the Planck scale and continues until spacetime phase transition occurs at a dynamical scale of about 1017 GeV. We show that during inflation, the initially large scale-invariant fluctuations reduce in amplitude to the appropriate magnitude suggested by tiny CMB anisotropies. The goal of this research is to derive the spectra of scalar fluctuations at the phase transition point, that is, the primordial spectra. A system of nonlinear evolution equations for the fluctuations is derived from the quantum gravity effective action. The running coupling constant is then expressed by a time-dependent average following the spirit of the mean field approximation. In this paper, we determine and examine various nonlinear terms, not treated in previous studies such as the exponential factor of the conformal mode. These contributions occur during the early stage of inflation when the amplitude is still large. Moreover, in order to verify their effects concretely, we numerically solve the evolution equation by making a simplification to extract the most contributing parts of the terms in comoving momentum space. The result indicates that they serve to maintain the initial scale invariance over a wide range beyond the comoving Planck scale. This is a challenge toward the derivation of the precise primordial spectra, and we expect in the future that it will lead to the resolution of the tensions that have arisen in cosmology.

Solitary deformation waves in two coaxial shells made of material with combined nonlinearity and forming the walls of annular and circular cross-section channels filled with viscous fluid

The aim of the paper is to obtain a system of nonlinear evolution equations for two coaxial cylindrical shells containing viscous fluid between them and in the inner shell, as well as numerical modeling of the propagation processes for nonlinear solitary longitudinal strain waves in these shells. The case when the stress-strain coupling law for the shell material has a hardening combined nonlinearity in the form of a function with fractional exponent and a quadratic function is considered. Methods. To formulate the problem of shell hydroelasticity, the Lagrangian–Eulerian approach for recording the equations of dynamics and boundary conditions is used. The multiscale perturbation method is applied to analyze the formulated problem. As a result of asymptotic analysis, a system of two evolution equations, which are generalized Schamel– Korteweg– de Vries equations, is obtained, and it is shown that, in general, the system requires numerical investigation. The new difference scheme obtained using the Grobner basis technique is proposed to discretize the system of evolution equations. Results. The exact solution of the system of evolution equations for the special case of no fluid in the inner shell is found. Numerical modeling has shown that in the absence of fluid in the inner shell, the solitary deformation waves have supersonic velocity. In addition, for the above case, it was found that the strain waves in the shells retain their velocity and amplitude after interaction, i.e., they are solitons. On the other hand, calculations have shown that in the presence of a viscous fluid in the inner shell, attenuation of strain solitons is observed, and their propagation velocity becomes subsonic.

Evolution of solitary hydroelastic strain waves in two coaxial cylindrical shells with the Schamel physical nonlinearity

The paper considers the formulation and solution of the hydroelasticity problem for studying wave processes in the system of two coaxial shells containing fluids in the annular gap between them and in the inner shell. We investigate the axisymmetric case for Kirchhoff–Lave type shells whose material obeys a physical law with a fractional exponent of the nonlinear term (Schamel nonlinearity). The dynamics of fluids in the shells is considered within the framework of the incompressible viscous Newtonian fluid model. The derivation of the Schamel nonlinear equations of shell dynamics makes it possible to develop a mathematical formulation of the problem, which includes the obtained equations, the dynamics equations of two shells, the fluid dynamics equations and the boundary conditions at the shell-fluid interfaces and at the flow symmetry axis. The asymptotic analysis of the problem is performed using perturbation techniques, and the system of two generalized Schamel equations is obtained. This system describes the evolution of nonlinear solitary hydroelastic strain waves in the coaxial shells filled with viscous fluids, taking into account the inertia of the fluid motion. In order to determine the fluid stress at the shell-fluid interfaces, we perform linearization of the fluid dynamics equations for fluids in the annular gap and in the inner shell. The linearized equations are solved by the iterative method. The inertial terms are excluded from the equations in the first iteration, while, in the second iteration, these are the values found in the first iteration. A numerical solution of the system of nonlinear evolution equations is obtained by applying a new difference scheme developed using the Gröbner basis technique. Computational experiments are performed to investigate the effect of fluid viscosity and the inertia of fluid motion in the shells on the wave process. In the absence of fluids in the inner shell, the results of calculations demonstrate that the strain waves in the shells during elastic interactions do not change their shape and amplitude, i.e., they are solitons. The presence of viscous fluid in the inner shell leads to attenuation of the wave process.

Instability and rupture of surfactant-laden bilayer thin liquid films.

The stability of surfactant-laden bilayer thin films, where the top layer is subject to van der Waals driven breakup, is of particular relevance to applications where one thin liquid layer is spread on another, such as film-forming firefighting foams and multilayer coatings. Although there has been much prior modeling work on the stability of thin liquid bilayers, additional physical effects and assumptions were incorporated in those studies, making it difficult to isolate the influence of surfactant on the rupture of the top layer. The present work addresses this issue through application of the lubrication approximation to derive a coupled system of nonlinear evolution equations describing the perturbations to the liquid-liquid and liquid-air interfaces and the surfactant interfacial concentrations. The surfactant is assumed to be insoluble and can be present at each interface. Linear stability analysis suggests, and nonlinear simulations confirm, that by using surfactant that adsorbs to both interfaces, the rupture time can be increased by an order of magnitude relative to the surfactant-free case. However, we find it crucial to have the right amount of surfactant to generate strongly stabilizing Marangoni stresses without reducing the interfacial tension too much. Nonlinear simulations and linear stability analysis provide insight into the mechanisms of the delayed rupture and show how the direction and strength of the Marangoni stresses strongly depend on the viscosity ratio of the layers. These results can help guide the choice and design of surfactants to achieve more effective firefighting foams and more stable liquid coatings.

EXACT SOLITON SOLUTIONS FOR CONFORMABLE FRACTIONAL SIX WAVE INTERACTION EQUATIONS BY THE ANSATZ METHOD

In this paper, a conformable fractional time derivative of order [Formula: see text] is considered in view of the Lax-pair of nonlinear operators to derive a fractional nonlinear evolution system of partial differential equations, called the Fractional-Six-Wave-Interaction-Equations, which is derived in terms of one temporal plus one and two spatial dimensions. Further, an ansatz consisting of linear combinations of hyperbolic functions with complex coefficients is utilized to obtain an infinite set of exact soliton solutions for this system. Certain numerical examples are introduced to show the effectiveness of the ansatz method in obtaining exact solutions for similar systems of nonlinear evolution equations.

A General Class of Differential Hemivariational Inequalities Systems in Reflexive Banach Spaces

We consider an abstract system consisting of the parabolic-type system of hemivariational inequalities (SHVI) along with the nonlinear system of evolution equations in the frame of the evolution triple of product spaces, which is called a system of differential hemivariational inequalities (SDHVI). A hybrid iterative system is proposed via the temporality semidiscrete technique on the basis of the Rothe rule and feedback iteration approach. Using the surjective theorem for pseudomonotonicity mappings and properties of the partial Clarke’s generalized subgradient mappings, we establish the existence and priori estimations for solutions to the approximate problem. Whenever studying the parabolic-type SHVI, the surjective theorem for pseudomonotonicity mappings, instead of the KKM theorems exploited by other authors in recent literature for a SHVI, guarantees the successful continuation of our demonstration. This overcomes the drawback of the KKM-based approach. Finally, via the limitation process for solutions to the hybrid iterative system, we derive the solvability of the SDHVI with no convexity of functions u↦fl(t,x,u),l=1,2 and no compact property of C0-semigroups eAl(t),l=1,2.

Modeling invasion patterns in the glioblastoma battlefield.

Glioblastoma is the most aggressive tumor of the central nervous system, due to its great infiltration capacity. Understanding the mechanisms that regulate the Glioblastoma invasion front is a major challenge with preeminent potential clinical relevances. In the infiltration front, the key features of tumor dynamics relate to biochemical and biomechanical aspects, which result in the extension of cellular protrusions known as tumor microtubes. The coordination of metalloproteases expression, extracellular matrix degradation, and integrin activity emerges as a leading mechanism that facilitates Glioblastoma expansion and infiltration in uncontaminated brain regions. We propose a novel multidisciplinary approach, based on in vivo experiments in Drosophila and mathematical models, that describes the dynamics of active and inactive integrins in relation to matrix metalloprotease concentration and tumor density at the Glioblastoma invasion front. The mathematical model is based on a non-linear system of evolution equations in which the mechanisms leading chemotaxis, haptotaxis, and front dynamics compete with the movement induced by the saturated flux in porous media. This approach is able to capture the relative influences of the involved agents and reproduce the formation of patterns, which drive tumor front evolution. These patterns have the value of providing biomarker information that is related to the direction of the dynamical evolution of the front and based on static measures of proteins in several tumor samples. Furthermore, we consider in our model biomechanical elements, like the tissue porosity, as indicators of the healthy tissue resistance to tumor progression.

Darboux transformation and exact solutions for a (2 + 1)-dimensional integrable system of nonlinear evolution equations

In this paper, starting from a spectral problem, we construct a [Formula: see text]-dimensional integrable system of nonlinear evolution equations. Based on the Lax pair, the recursion operator and Darboux transformation for the whole hierarchy were constructed. As an application, some exact solutions for the hierarchy are obtained by using the Darboux transformation.

Diagnostics for eddy viscosity models of turbulence including data-driven/neural network based parameterizations

Classical eddy viscosity models add a viscosity term with turbulent viscosity coefficient whose specification varies from model to model. Turbulent viscosity coefficient approximations of unknown accuracy are typically constructed by solving associated systems of nonlinear evolution equations or by data driven approaches such as deep neural networks. Often eddy viscosity models over-diffuse, so additional fixes are added. This process increases model complexity and decreases model comprehensibility, leading to the following two questions: Is an eddy viscosity model needed? Does the eddy viscosity model fail? This report derives diagnostic quantities of interest that answer these two questions. A notable quality of the derived quantities of interest for the eddy viscosity model is that they are a posteriori computable and require no a priori knowledge of the parameterization. For neural network based parameterizations these diagnostic quantities provide an indication of when the eddy viscosity model fails due to over diffusion of the flow.

Algorithm for Solving the Cauchy Problem for One Infinite-Dimensional System of Nonlinear Differential Equations

The Cauchy problem for an infinite-dimensional system of nonlinear evolution equations, which is a generalization of the Langmuir chain, is considered. The global solvability of the problem in the class of rapidly decreasing functions is established. By the inverse spectral method, an algorithm for constructing the solution is obtained.

Time-Dependent Polynomials with One Double Root, and Related New Solvable Systems of Nonlinear Evolution Equations

Recently new solvable systems of nonlinear evolution equations—including ODEs, PDEs and systems with discrete time—have been introduced. These findings are based on certain convenient formulas expressing the kth time-derivative of a root of a time-dependent monic polynomial in terms of the kth time-derivative of the coefficients of the same polynomial and of the roots of the same polynomial as well as their time-derivatives of order less than k. These findings were restricted to the case of generic polynomials without any multiple root. In this paper some of these findings—those for $$k=1$$ and $$k=2$$ —are extended to polynomials featuring one double root; and a few representative examples are reported of new solvable systems of nonlinear evolution equations.

Adaptive high-order splitting methods for systems of nonlinear evolution equations with periodic boundary conditions

We assess the applicability and efficiency of time-adaptive high-order splitting methods applied for the numerical solution of (systems of) nonlinear parabolic problems under periodic boundary conditions. We discuss in particular several applications generating intricate patterns and displaying nonsmooth solution dynamics. First, we give a general error analysis for splitting methods for parabolic problems under periodic boundary conditions and derive the necessary smoothness requirements on the exact solution in particular for the Gray–Scott equation and the Van der Pol equation. Numerical examples demonstrate the convergence of the methods and serve to compare the efficiency of different time-adaptive splitting schemes and of splitting into either two or three operators, based on appropriately constructed a posteriori local error estimators.

Identification problems for systems of nonlinear evolution equations and functional equations

In this work, we consider two inverse problems for systems of nonlinear differential and differential-difference evolution equations. We propose a new technique for the solution of such inverse problems. We reduce the problem to a system of functional equations by using the given data and then we obtain the solution by the method of Kuczma (Functional Equations in a Single Variable, 1968).

Exact traveling wave solutions for system of nonlinear evolution equations.

In this work, recently deduced generalized Kudryashov method is applied to the variant Boussinesq equations, and the (2 + 1)-dimensional breaking soliton equations. As a result a range of qualitative explicit exact traveling wave solutions are deduced for these equations, which motivates us to develop, in the near future, a new approach to obtain unsteady solutions of autonomous nonlinear evolution equations those arise in mathematical physics and engineering fields. It is uncomplicated to extend this method to higher-order nonlinear evolution equations in mathematical physics. And it should be possible to apply the same method to nonlinear evolution equations having more general forms of nonlinearities by utilizing the traveling wave hypothesis.

Harmonic Maps Surfaces and Relativistic Strings

The harmonic map is introduced and several physical applications are presented. The classical nonlinear σ model can be looked at as the embedding of a two-dimensional surface in a threedimensional sphere, which is itself embedded in a four-dimensional space. A system of nonlinear evolution equations are obtained by working out the zero curvature condition for the Gauss equations relevant to this geometric formulation.

On the impact deformation of an incompressible half-space under the action of a shear load of variable direction

Using the example of a one-dimensional planar problem for a nonlinear elastic incompressible half-space, the loading process is considered in which the shear action on the boundary plane changes with respect to both intensity and direction. It is shown that, in the regions of the space where the nonlinearity of the medium becomes a significant factor, the solution in the nearfront region of the shock wave is determined by a system of nonlinear evolution equations. The general solution of the evolution system is obtained. As an example, a partial solution of the system is considered for one of the simplest boundary conditions. A parametric method is presented for determining the displacements on the basis of solution of the evolution system.

The modified simple equation method and its applications to (2+1)-dimensional systems of nonlinear evolution equations

The modified simple equation method is employed to find the exact traveling wave solutions involving parameters for nonlinear systems of evolution equations via the (2+1)-dimensional Konopelchneko-Dubrovsky equations and the (2+1)-dimensional Nizhnik-Novikov-Vesselov equations in two dimensions. When these parameters are taken to be special values, the solitary wave solutions are derived from the exact traveling wave solutions. It is shown that the modified simple equation method provides an effective and a more powerful mathematical tool for solving nonlinear evolution equations in mathematical physics. Comparison between our results and the well-known results will be presented. Key words: Modified simple equation method, Konopelchneko-Dubrovsky equations, Nizhnik-Novikov-Vesselov equations, exact traveling solutions, solitary wave solutions.

Evolution of deep-water waves under wind forcing and wave breaking effects: Numerical simulations and experimental assessment

The evolution of two-dimensional dispersive focusing wave groups in deep water under wind forcing and wave breaking effects is investigated numerically and measurements collected from wind–wave experiments are used to evaluate the numerical simulations. Wind forcing is modeled by introducing into the dynamic boundary condition a surface slope coherent pressure distribution, which is expressed through Miles’ shear instability theory and Jeffreys’ sheltering model. To activate Jeffreys’ model in simulating waves evolving under wind forcing, an air flow separation criterion depending on wind speed and wave steepness is proposed. Direct comparisons of the measurements and the simulations are made by including the wind-driven current in the simulations. To simulate breaking waves, an eddy viscosity model is incorporated into a system of nonlinear evolution equations to dissipate wave energy and to predict surface elevation after breaking. For wave groups under no wind action, the eddy viscosity model simulates well the energy dissipation in breaking waves and predicts well the surface elevation after breaking. Under the weaker wind forcing condition, after consideration of the wind-driven current, the numerical model produces satisfying predictions. As the wind forcing becomes stronger, the disparity between the experiments and the simulations becomes more evident while the numerical results are still regarded as acceptable. The relative importances of the Miles’ and the Jeffreys’ models for waves under wind forcing are discussed through additional numerical tests.

Nonlinear dynamics of an optical parametric exciton-polariton oscillator in semiconductor microcavity

The dynamics of optical parametric exciton-polariton oscillations in semiconductor microcavities was studied, taking into account the conversion processes of the two pump polaritons into signal and idler polaritons and vice versa. The system of nonlinear evolution equations was obtained for describing the time evolution for the densities of pump, signal and idler polaritons. The periodic and aperiodic regimes of the evolution of polariton densities exist. The period and amplitude of oscillations of the polariton densities essentially depend on the initial densities of polaritons, initial phase difference and resonance detuning. For the definite relation between the parameters, the aperiodic evolution of the system is possible, too, representing the conversion of only part of the pump polaritons, which finishes the evolution of the system. The significant dependence of the period and amplitude of polariton oscillations on the initial phase difference is evidence of the possibility of the phase controlling of the polariton dynamics. The damping of the system of coherent polaritons leads to the oscillatory decay of polariton density in time, the rate of which is defined by the polariton damping constants.

Invariant subspaces and exact solutions of a class of dispersive evolution equations

The invariant subspace method is used to classify a class of systems of nonlinear dispersive evolution equations and determine their invariant subspaces and exact solutions. A crucial step is to take subspaces of solutions to linear ordinary differential equations as invariant subspaces that systems of evolution equations admit. A few examples of presenting exact solutions with generalized separated variables illustrate the effectiveness of the invariant subspace method in solving systems of nonlinear evolution equations.