Dissipative Equations Research Articles

Multicomponent second-order dissipative relativistic hydrodynamics with binary reactive collisions

We derive the multi-component second-order dissipative relativistic hydrodynamic equations using the moment-expansion method. By computing the transport coefficients using hard-sphere interactions, we investigate the role of multiple components and the reactive collisions. We find that both of these factors increase the effective cross-section and hence decrease the transport coefficients and relaxation times. We further compute such transport properties using leading-order perturbative QCD cross-sections. For both types of cross-sections, we find that the ratio between vector current relaxation time and conductivity for a multi-component fluid is notably different from that for a single-component fluid. Therefore, the current study provides a more applicable guideline for such a ratio in phenomenological hydrodynamics simulations.

On fractional symmetry group scheme to the higher-dimensional space and time fractional dissipative Burgers equation

In this paper, we studied a higher-dimensional space and time fractional model, namely, the (3+1)-dimensional dissipative Burgers equation which can be used to describe the shallow water waves phenomena. Here, the analyzed tool is the Lie symmetry scheme in the sense of the Riemann–Liouville fractional derivative. First of all, the symmetry of this considered equation was yielded. Then, based on the above obtained symmetry, the one-parameter Lie group was obtained. Subsequently, this model can be changed into the lower-dimensional equation with the Erdélyi–Kober fractional operators. Lastly, conservation laws of this studied equation via a new conservation theorem were also received. After such a series of processing, these new results play an important role in our understanding of this higher-dimensional space and time differential equations.

Singularity formation in the deterministic and stochastic fractional Burgers equation

This study is motivated by the question of how singularity formation and other forms of extreme behavior in nonlinear dissipative partial differential equations are affected by stochastic excitations. To address this question we consider the 1D fractional Burgers equation with additive colored noise as a model problem. This system is interesting, because in the deterministic setting it exhibits finite-time blow-up or a globally well-posed behavior depending on the value of the fractional dissipation exponent. The problem is studied by performing a series of accurate numerical computations combining spectrally-accurate spatial discretization with a Monte-Carlo approach. First, we carefully document the singularity formation in the deterministic system in the supercritical regime where the blow-up time is shown to be a decreasing function of the fractional dissipation exponent. Our main result for the stochastic problem is that there is no evidence for additive noise to regularize the evolution by suppressing singularities in the supercritical regime, or for the noise to trigger singularities in the subcritical regime. However, as the noise amplitude becomes large, the maximum existence time of the solution in the supercritical regime is shown to exhibit an increasingly non-Gaussian behavior. Analogous observations are also made for the maximum attained values of the enstrophy and the times when the maxima occur in the subcritical regime.

Data-driven reduced-order modeling of spatiotemporal chaos with neural ordinary differential equations.

Dissipative partial differential equations that exhibit chaotic dynamics tend to evolve to attractors that exist on finite-dimensional manifolds. We present a data-driven reduced-order modeling method that capitalizes on this fact by finding a coordinate representation for this manifold and then a system of ordinary differential equations (ODEs) describing the dynamics in this coordinate system. The manifold coordinates are discovered using an undercomplete autoencoder-a neural network (NN) that reduces and then expands dimension. Then, the ODE, in these coordinates, is determined by a NN using the neural ODE framework. Both of these steps only require snapshots of data to learn a model, and the data can be widely and/or unevenly spaced. Time-derivative information is not needed. We apply this framework to the Kuramoto-Sivashinsky equation for domain sizes that exhibit chaotic dynamics with again estimated manifold dimensions ranging from 8 to 28. With this system, we find that dimension reduction improves performance relative to predictions in the ambient space, where artifacts arise. Then, with the low-dimensional model, we vary the training data spacing and find excellent short- and long-time statistical recreation of the true dynamics for widely spaced data (spacing of ∼ 0.7 Lyapunov times). We end by comparing performance with various degrees of dimension reduction and find a "sweet spot" in terms of performance vs dimension.

High-order Numerical Homogenization for Dissipative Ordinary Differential Equations

We propose a high order numerical homogenization method for dissipative ordinary differential equations (ODEs) containing two time scales. Essentially, only first order homogenized model globally in time can be derived. To achieve a high order method, we have to adopt a numerical approach in the framework of the heterogeneous multiscale method (HMM). By a successively refined microscopic solver, the accuracy improvement up to arbitrary order is attained providing input data smooth enough. Based on the formulation of the high order microscopic solver we derived, an iterative formula to calculate the microscopic solver is then proposed. Using the iterative formula, we develop an implementation to the method in an efficient way for practical applications. Several numerical examples are presented to validate the new models and numerical methods.

Fixed analytic radius lower bound for the dissipative KdV equation on the real line

Fixed analytic radius lower bound for the dissipative KdV equation on the real line

Consolidation of partially saturated ground improved by impervious column inclusion: Governing equations and semi-analytical solutions

This study focuses on the consolidation behavior and mathematical interpretation of partially-saturated ground improved by impervious column inclusion. The constitutive relations for soil skeleton, pore air and pore water for partially saturated soils are proposed in the context of partially-saturated ground improved by impervious column inclusion. Settlement equation and dissipation equations of excess pore air/water pressures for a partially saturated improved ground are then derived. The semi-analytical solutions for ground settlement and pore pressure dissipation are then obtained through the Laplace transform and validated by the existing solutions for two special cases in the literature and the numerical results obtained from the finite difference method. A series of parametric studies is finally conducted to investigate the influence of some key factors on consolidation of partially saturated ground improved by impervious column inclusion. Based on the parametric study, it can be found that a higher value of the area replacement ratio or modulus of the pile results in a longer dissipation time of excess pore air pressure (PAP), a shorter dissipation time of excess pore water pressure (PWP), and a lower normalized settlement.

Stability of high order finite difference and local discontinuous Galerkin schemes with explicit-implicit-null time-marching for high order dissipative and dispersive equations

Time discretization is an important issue for time-dependent partial differential equations (PDEs). For the k-th (k≥2) order PDEs, the explicit time-marching method may suffer from a severe time step restriction τ=O(hk) for stability. The implicit and implicit-explicit (IMEX) time-marching methods can overcome this constraint. However, for the equations with nonlinear high derivative terms, the IMEX methods are not good choices either, since a nonlinear algebraic system must be solved (e.g. by Newton iteration) at each time step. The explicit-implicit-null (EIN) time-marching method is designed to cope with the above mentioned shortcomings. The basic idea of the EIN method discussed in this paper is to add and subtract a sufficiently large linear highest derivative term on one side of the considered equation, and then apply the IMEX time-marching method to the equivalent equation. The EIN method so designed does not need any nonlinear iterative solver, and the severe time step restriction for explicit methods can be removed. Coupled with the EIN time-marching method, we will discuss the high order finite difference and local discontinuous Galerkin schemes for solving high order dissipative and dispersive equations, respectively. By the aid of the Fourier method, we perform stability analysis for the schemes on the simplified equations with periodic boundary conditions, which demonstrates the stability criteria for the resulting schemes. Even though the analysis is only performed on the simplified equations, numerical experiments show that the proposed schemes are stable and can achieve optimal orders of accuracy for both one-dimensional and two-dimensional linear and nonlinear equations.

Formation of acoustic nonlinear structures in non-Maxwellian trapping plasmas

In this paper, expressions of number densities for electron trapping for generalized (r, q), kappa, and Cairns distribution functions, respectively, are reported using the approach adopted by Landau and Lifshitz for Maxwellian trapping of electrons. For illustrative purposes, dispersive and dissipative equations for ion-acoustic waves are obtained in the presence of non-Maxwellian trapped electrons in the small amplitude limit. The solutions of the modified dispersive and dissipative nonlinear equations are reported, and a graphical analysis is given to present a detailed comparison of non-Maxwellian and Maxwellian trapping. The results presented here, to the best of authors' knowledge, are a first attempt of this kind. It is expected that the present investigation will unravel new horizons for future research and encourage the researchers to search for the nonlinear structures presented in this paper in the satellite data.

An energy-preserving computational approach for the semilinear space fractional damped Klein–Gordon equation with a generalized scalar potential

Naturally preserving dissipative or conservative structures of a given continuous system in discrete analogs is of high demand when designing numerical schemes for dissipative or Hamiltonian partial differential equations. Armed with this fact, the proposed computational study focuses on the issue of considering energy-preserving discrete numerical schemes for the Riesz space-fractional Klein-Gordon equation with a generalized scalar potential. For the sake of more clearness, a combined numerical scheme that owes energy-preserving properties is a target to be achieved here. This is done by adapting Galerkin spectral approximation based on Legendre polynomials for Riesz space Laplacian operator side by side to invoke a Cranck–Nickolson scheme in time direction after making order reduction of the original system and a special second-order approximation of the scalar potential derivative. The numerical properties (stability, uniqueness, and convergence) of the scheme are well investigated. Numerical simulations also show that the proposed scheme can inherit the physical properties as the original problem and the numerical solution is stable and convergent to the exact solution.

The Soliton Wave Solutions and Bifurcations of the (2 + 1)-Dimensional Dissipative Long Wave Equation

With the help of the bifurcation theory of dynamical differential system and maple software, we shall devote ourselves to research travelling wave solutions and bifurcations of the (2 + 1)-dimensional dissipative long wave equation. The study of travelling wave solutions for long wave equation derives a planar Hamiltonian system. Based on phase portraits, we obtain exact explicit expressions of some bounded traveling wave solutions and some important singular traveling wave solutions, under different parametric conditions.

Observability, controllability and stability of a nonlinear RLC circuit in form of a Duffing oscillator by means of theoretical mechanical approach

Abstract In this research article, observability, controllability and stability of a nonlinear RLC circuit with a nonlinear capacitor is investigated as a Duffing oscillator beginning with the dissipative equations of generalized motion using Lagrange-dissipative model ({L, D} -model briefly). The force related to the potential energy, equilibria, and their well known stability properties are given using state space approach. Prerequisite that the condition for a Legendre transform is fulfilled, for the same system, also Hamiltonian of the system is found. Using Hamiltonian and dissipation function, dissipative canonical equations are obtained. These equations are written in state space form. Then the equality to the same results obtained using the dissipative equations of generalized motion related equilibria and their stability was shown. Thus a Lyapunov function as residual energy function (REF) is justified in terms of stability of the overall system. As last step, different electrical and mechanical (physical) realization possibilities are discussed.

Contribution to the definition of a simple focusing test case and ONERA results for the two test cases of the sonic boom focusing predictions special session

For usual cruise flight conditions, the prediction of the sonic boom on ground caused by an aircraft flying supersonically can be predicted numerically by ray tracing codes according to the geometrical acoustics hypotheses. Such codes account for the non-linear and dissipative (thermoviscous and molecular relaxation) effects that occur during the propagation from flight level down to the ground by solving dissipative Burgers' equations along the rays, accounting for the evolution of the ray tube area. Such an approach is not applicable to predict the pressure waveform in the neighbourhood of a caustics that can be generated by the cusp of the focussing pressure wave generated under specific unsteady flight conditions of the aircraft. For such focussing conditions, the pressure waveform can be modelled by the Tricomi equations. The BANGV code has been used to predict the pressure waveforms in focussing conditions of the sonic boom focusing predictions special session of the 182nd ASA meeting. The results obtained for the two test-cases will be presented in this paper to be compared to other participants' results and contribute to the validation of numerical simulation methods of focussing boom.

Radiative thermalization in semiclassical simulations of light-matter interaction

Prediction of the equilibrium populations in quantum dynamics simulations of molecules exposed to black-body radiation has proved challenging for semiclassical treatments, with the usual Ehrenfest and Maxwell-Bloch methods exhibiting serious failures. In this context, we explore the behavior of a recently introduced semiclassical model of light-matter interaction derived from a dissipative Lagrangian [C. M. Bustamante, E. D. Gadea, A. Horsfield, T. N. Todorov, M. C. Gonz\'alez Lebrero, and D. A. Scherlis, Phys. Rev. Lett. 126, 087401 (2021)]. It is shown that this model reproduces the Boltzmann populations for two-level systems, predicting the black-body spectra in approximate agreement with Planck's distribution. In multilevel systems, small deviations from the expected occupations are seen beyond the first excited level. By averaging over fast oscillations, a rate equation is derived from the dissipative equation of motion that makes it possible to rationalize these deviations. Importantly, it enables us to conclude that this model will produce the correct equilibrium populations provided the occupations of the lowest levels remain close to unity, a condition satisfied at low temperature or small excitations.

A hybrid computational method for local fractional dissipative and damped wave equations in fractal media

In this paper, we propose a copulation of the local fractional variational iteration technique and local fractional natural transform (LFNT) for solving the dissipative and damped wave equations with local fractional derivatives. The convergence analysis for the solution obtained through this newly suggested method is also discussed. The numerical simulations for computed solutions are presented on the Cantor set. The computational procedure shows that the newly proposed local fractional method pertaining to the LFNT is very easy and effective to generate solutions for given local fractional wave equations (LFWEs). Moreover, the generated solutions are also in nice correspondence with the previous results acquired by other existing techniques.

Well-posedness of solutions for the dissipative Boussinesq equation with logarithmic nonlinearity

In this paper, we consider a class of Boussinesq-type equations with logarithmic nonlinearity. By employing the classical Faedo–Galerkin method, we first establish the local well-posedness of solutions. Then we investigate the dynamical behaviors of solutions. More precisely, for the solutions with subcritical or critical initial energy, we prove that they exist globally and are uniformly bounded when I(u0)>0, where I(u0) denotes the Nehari functional with the initial value u0. Moreover, under further appropriate assumptions about the initial data, we derive the exponential energy decay estimates of global solutions. In particular, for the solutions with subcritical or critical initial energy, we show that they can be extended over time (the whole half line) and then blow up at infinite time when I(u0)<0. Last but not least, by developing some new methods, we prove the existence of infinite time blow-up solutions with arbitrary high initial energy.

Photoluminescence of Cis-Polyacetylene Semiconductor Material

Photoluminescence (PL) is one of the key experimental characterizations of optoelectronic materials, including conjugated polymers (CPs). In this study, a simplified model of an undoped cis-polyacetylene (cis-PA) oligomer was selected and used to explain the mechanism of photoluminescence (PL) of the CPs. Using a combination of the ab initio electronic structure and a time-dependent density matrix methodology, the photo-induced time-dependent excited state dynamics were computed. We explored the phonon-induced relaxation of the photoexcited state for a single oligomer of cis-PA. Here, the dissipative Redfield equation of the motion was used to compute the dissipative excited state dynamics of electronic degrees of freedom. This equation used the nonadiabatic couplings as parameters. The computed excited state dynamics showed that the relaxation rate of the electron is faster than the relaxation rate of the hole. The dissipative excited-state dynamics were combined with radiative recombination channels to predict the PL spectrum. The simulated results showed that the absorption and emission spectra both have a similar transition. The main result is that the computed PL spectrum demonstrates two mechanisms of light emission originating from (i) the inter-band transitions, corresponding to the same range of transition energies as the absorption spectrum and (ii) intra-band transitions not available in the absorption spectra. However, the dissipative Redfield equation of the motion was used to compute the electronic degrees of freedom of the nonadiabatic couplings, which helped to process the time propagation of the excited dynamic state. This excited dynamic state shows that the relaxation rate of the electron is faster than the relaxation rate of the hole, which can be used for improving organic semiconductor materials for photovoltaic and LED applications.

A novel energy‐based framework for characterizing the strain‐softening behavior of CFRP composites using cyclic loading

AbstractComplex failure modes of carbon fiber reinforced polymer (CFRP) composites create significant challenges for CFRP components' progressive damage analysis (PDA). Though important in PDA, the phenomenological strain‐softening model (SSM) for characterizing the degradation behavior in the fiber‐parallel direction (dominated by fiber breakage) of CFRP is difficult to be constructed by current approaches due to the poor toughness of carbon fibers. The work aims to obtain a refined SSM of plain woven CFRP composites through experimental methods and apply it to the PDA of corresponding components. A novel phenomenological cyclic‐loading‐based strain‐softening model (CLSSM) was proposed based on the theory of equivalent dissipation energy during static and cyclic loading. Firstly, the release behavior of dissipation energy during the cyclic loading process was researched. Secondly, a newly designed search algorithm taking degraded mechanical properties during cyclic loading as input and taking the static strain‐softening curve as output was utilized to build up CLSSM. Thirdly, a three‐point bending test was conducted and a corresponding numerical model was constructed for comparison to verify the proposed model. The entire framework is highly programmatic, and results show that CLSSM is fast for characterizing the strain‐softening behavior, and effectively simulates the progressive failure process of CFRP components.

Time periodic solutions to the 2D quasi-geostrophic equation with the supercritical dissipation

We consider the 2D dissipative quasi-geostrophic equation with the time periodic external force and prove the existence of a unique time periodic solution in the case of the supercritical dissipation. In this case, the smoothing effect of the semigroup generated by the dissipation term is too weak to control the nonlinearity in the Duhamel term of the corresponding integral equation. In this paper, we give a new approach which does not depend on the contraction mapping principle for the integral equation.

Explicit solutions to the time-fractional generalized dissipative Kawahara equation

In this paper, the generalized dissipative Kawahara equation in the sense of conformable fractional derivative is presented and solved by applying the tanh-coth-expansion and sine-cosine function techniques. The quadratic-case and cubic-case are investigated for the proposed model. Expected solutions are obtained with highlighting to the effect of the presence of the alternative fractional-derivative and the effect of the added dissipation term to the generalized Kawahara equation. Some graphical analysis is presented to support the findings of the paper. Finally, we believe that the obtained results in this work will be important and valuable in nonlinear sciences and ocean engineering.