Evolution Equations Research Articles

Hot leptogenesis

We investigate a class of leptogenesis scenarios in which the sector containing the lightest right-handed neutrino establishes kinetic equilibrium at a temperature TN1 > TSM, where TSM is the temperature of the Standard Model sector. We study the reheating processes which realise this “hot leptogenesis” and the conditions under which kinetic and chemical equilibrium can be maintained. We derive and solve two sets of evolution equations, depending on the presence of chemical equilibrium within the hot sector, and numerically solve these for benchmark scenarios. We compare the viable parameter space of this model with standard leptogenesis scenarios with a thermal initial condition and find that hot leptogenesis resolves the neutrino and Higgs mass fine-tuning problems present in the standard scenario.

One-dimensional coefficient inverse problems by transformation operators

Abstract We prove the uniqueness for an inverse problem of determining a matrix coefficient P(x) of a system of evolution equations σ ∂ t u = ∂ x 2 u ( t , x ) − P ( x ) u ( t , x ) for 0 < x < ℓ and 0 < t < T , where σ ∈ C ∖ { 0 } , ℓ > 0 and T > 0 are arbitrarily given and P is real-valued. The uniqueness results assert that two solutions have the same Cauchy data at x = 0 over ( 0 , T ) and the same initial value or the final value which is positive on [ 0 , ℓ ] , then the zeroth-order coefficient is uniquely determined on [ 0 , ℓ ] . The uniqueness for inverse coefficient problem for a system of evolution equations without boundary conditions over the whole boundary is an open problem even in the one-dimension in the case where only initial value is given as spatial data. Moreover, in the case of the zero initial condition, we prove the uniqueness in the half of the spatial interval.

Stochastic wavevector model for rapidly distorted compressible turbulence

A stochastic wavevector approach is formulated to accurately represent compressible turbulence subject to rapid deformations. This approach is inspired by the incompressible particle representation model of Kassinos & Reynolds (1994), and preserves the exact nature of compressible rapid distortion theory (RDT). The adoption of a stochastic – rather than Fourier – perspective simplifies the transformation of statistics to physical space and serves as a starting point for the development of practical turbulence models. We assume small density fluctuations and isentropic flow to obtain a transport equation for the pressure fluctuation. This results in four fewer transport equations compared with the compressible RDT model of Yu & Girimaji (Phys. Fluids, vol. 19, 2007, 041702). The final formulation is closed in spectral space and only requires numerical approximation for the transformation integrals. The use of Monte Carlo for unit wavevector integration motivates the representation of the moments as stochastic variables. Consistency between the Fourier and stochastic representation is demonstrated by showing equivalency between the evolution equations for the velocity spectrum tensor in both representations. Sample clustering with respect to orientation allows for different techniques to be used for the wavevector magnitude integration. The performance of the stochastic model is evaluated for axially compressed turbulence, serving as a simplified model for shock–turbulence interaction, and is compared with linear interaction approximations and direct numerical simulation (DNS). Pure and compressed sheared turbulence at different distortion Mach numbers are also computed and compared with RDT/DNS data. Finally, two additional deformations are applied and compared with solenoidal and pressure-released limits to demonstrate the modelling capability for generic rapid deformations.

Updating TMD parton densities in a proton within the Kimber-Martin-Ryskin approach

We present analytical expressions for the transverse momentum dependent (unintegrated) gluon and quark densities in a proton derived at leading order of QCD running coupling and valid at both small and large x. The calculations are performed using the Kimber-Martin-Ryskin/Watt-Martin-Ryskin prescription (in both differential and integral formulations) with different treatment of kinematical constraint, which reflects the angular and strong ordering conditions for parton emissions. As an input, the analytical solution of QCD evolution equations for conventional (collinear) parton distributions is applied, where the valence and nonsinglet quark parts obey the Gross-Llewellyn-Smith and Gottfried sum rules, and momentum conservation for the singlet quark and gluon densities is taken into account. Several phenomenological parameters are extracted from the combined fit to precision data on the proton structure function F2(x,Q2) collected by the BCDMS, H1, and ZEUS Collaborations, comprising a total of 933 points from 5 datasets. Comparison with the numerical results obtained by other groups is presented, and the phenomenological application to the inclusive b-jet production at the LHC is given. Published by the American Physical Society 2025

Network and Phase Symmetries Reveal That Amplitude Dynamics Stabilize Decoupled Oscillator Clusters

Oscillator networks display intricate synchronization patterns. Determining their stability typically requires incorporating the symmetries of the network coupling. Going beyond analyses that appeal only to a network’s automorphism group, we explore synchronization patterns that emerge from the phase-shift invariance of the dynamical equations and symmetries in the nodes. We show that these nonstructural symmetries simplify stability calculations. We analyze a ring-network of phase–amplitude oscillators that exhibits a “decoupled” state in which physically-coupled nodes appear to act independently due to emergent cancellations in the equations of dynamical evolution. We establish that this state can be linearly stable for a ring of phase–amplitude oscillators, but not for a ring of phase-only oscillators that otherwise require explicit long-range, nonpairwise, or nonphase coupling. In short, amplitude–phase interactions are key to stable synchronization at a distance.

Stability of relativistic spheres powered by energy profile in Einstein Gauss–Bonnet gravity

In this paper, we study the dynamical irregularity of the locally anisotropic spherical fluids in the context of Einstein–Gauss–Bonnet theory. We aim to describe the causes of energy-density irregularity of self-gravitating fluids and explain how those causes evolved from a homogeneous distribution at first. After computing field equations, we formulate two independent components of evolution equations. This expression involves the Weyl tensor and dynamical variables that would lead us to explain the emergence of inhomogeneity patterns. The relevant quantities involved in the irregularities within the initially homogeneous system are analyzed by considering particular nondissipative and dissipative distribution cases. We find the theoretical irregularity factor consistent with astrophysical observations. With this relation, it is explicitly demonstrated that in the presence of Einstein–Gauss–Bonnet gravity terms, the inhomogeneity term decreases its role gradually as the observer moves away from the center towards the boundary surface.

Solitary wave solutions and stability analysis of the fractional Sawada-Kotera equation using the extended modified auxiliary equation mapping method

Abstract In this study, we investigate wave solutions to the space–time fractional (1 + 1)-dimensional Sawada-Kotera equation, a nonlinear integrable evolution equation with noteworthy applications in shallow water waves and other fluid systems utilizing the extended modified auxiliary equation mapping method, an adaptable and specific analytical technique. We establish several novel solitary wave solutions, including trigonometric, rational, exponential, and hybrid solitary wave solutions to the stated model under two different sets of parameters. The solutions include a variety of waveforms such as kink, pulse, general, and other solitons, which hold numerous applications in nonlinear physics and engineering. We examine the effects of the beta fractional parameter on the obtained solitons analytically and graphically. The stability of these solutions under system parameters is also assessed. Graphical representations of the solutions are provided in 2D, contour, and 3D formats using MATLAB with appropriately selected parameter values. The method proves to be effective and efficient for explaining the fractional nonlinear model, demonstrating its potential for addressing the space–time fractional Sawada-Kotera equation.

Field-dependent diffeomorphisms and the transformation of surface charges between gauges

When studying gauge theories in the presence of boundaries, local symmetry transformations are typically classified as gauge or physical depending on whether the associated charges vanish or not. Here, we propose that physical charges should further be refined into “dynamical” or “kinematical” depending on whether they are associated with evolution equations or not. To support this proposal, we analyze (A)dS3 gravity with boundary Weyl rescalings and compare the solution spaces in Bondi-Sachs and Fefferman- Graham coordinates. In line with previous accounts, we show that the Weyl charge vanishes in the Bondi-Sachs gauge but not in the Fefferman-Graham gauge. Conversely, the charges arising from the metric Chern-Simons Lagrangian behave in the opposite way. This indicates that the gauge-dependent Weyl charge differs fundamentally from charges like mass and angular momentum. This interpretation is reinforced by two key observations: the Weyl conformal factor does not satisfy any evolution equation, and the associated charge arises from a corner term in the symplectic structure. These properties justify assigning the Weyl charge a kinematical status. These results can also be derived using the field-dependent diffeomorphism that maps between the two gauges. Importantly, this diffeomorphism does not act tensorially on the variational bi-complex due to its field dependency, and is able to “toggle” charges on or off. This provides an example of a large diffeomorphism between gauges, as opposed to a residual diffeomorphism within a gauge.

Late time phenomena in f(T,T) gravity framework: role of H0 priors

This study explored the behavior of the f(T,T) cosmological model with the use of various data set combinations. We also compared the results for this model between the Pantheon+ (without SH0ES) and the Pantheon+ &SH0ES (with SH0ES) data sets. Additionally, we incorporated data from BAO along with H0 priors. We observed that integrating SH0ES data points leads to a higher estimation of H0 than Pantheon+ (without SH0ES). We perform an extensive MCMC analysis for each combination of data sets, providing constraints on the model parameters. We also computed the χmin2 value for each combination of data sets to evaluate the chosen model against the standard ΛCDM model. Our primary finding is that the various dataset combinations in the f(T,T) model we examined relate to a range of Hubble constants, which could contribute to reducing the cosmic tension associated with this parameter. Additionally, we investigate the evolution of matter fluctuations by solving the density contrast evolution equation numerically. We calculate numerical solutions for the weighted growth rate fσ8 using these findings. We plotted the cosmological background parameters to check the behavior of the f(T,T) model in late-time. Based on the behavior of these background cosmological parameters, we conclude that our selected models reflect the late-time cosmic dynamics of the Universe.

On mapping moving curves to complex KdV-type equations for the Darboux triad in Euclidean space

In this paper, we show a class of moving curves for the Darboux triad in Euclidean space that can be mapped onto complex KdV-type equations via the generalization of the Heisenberg ferromagnetic spin chain model. Furthermore, we give an approximation of the relationships with certain standard evolution equations.

Fractional integrable Toda lattice and hierarchy

A fractional extension of the integrable Toda lattice with decaying data on the line is obtained. Completeness of squared eigenfunctions of a linear discrete real tridiagonal eigenvalue problem is derived. This completeness relation allows nonlinear evolution equations expressed in terms of operators to be written in terms of underlying squared eigenfunctions and is related to a discretization of the continuous Schrödinger equation. The methods are discrete counterparts of continuous ones recently used to find fractional integrable extensions of the Korteweg–de Vries (KdV) and nonlinear Schrödinger (NLS) equations. Inverse scattering transform (IST) methods are used to linearize and find explicit soliton solutions to the integrable fractional Toda (fToda) lattice equation. The methodology can also be used to find and solve fractional extensions of a Toda lattice hierarchy.

Abundant Explicit Non-Traveling Wave Solutions to the \((3 + 1)\)-Dimensional Nonlinear Evolution Equation Using a Generalized Variable Separation Method

Non-traveling wave solutions allow us to characterize more complex natural phenomena across various scientific fields, which may not be easily analyzed using soliton solutions. This study introduces a novel approach for deriving a wide variety of explicit non-traveling wave solutions to the (3 + 1)-dimensional nonlinear evolution equation, utilizing a modified generalized variable separation technique. The newly obtained solutions encompass different non-traveling forms, including periodic solitary waves and soliton-like structures. These results underscore the efficacy of the method in tackling complex nonlinear partial differential equations. The findings presented in this article constitute innovative contributions to the equation modeling the velocity of water waves on the surface of shallow water. Furthermore, our employed technique can also provide a foundation for investigating the stability, dynamics, and practical applications of solutions to other similar problems.

Efficient time-stepping for the evolution equations of damage-induced growth and remodelling in soft biological tissues

Efficient time-stepping for the evolution equations of damage-induced growth and remodelling in soft biological tissues

Spectral criteria for the asymptotic constancy of solutions to implicit difference equations

This paper deals with spectral criteria for the asymptotic constancy of solutions to the implicit difference equation Cx ( n + 1 ) = Tx ( n ) + y ( n ) in a Banach space X , where the bounded sequence { y ( n ) } n is asymptotically constant. The main result states that, if 1 is either not in σ Γ ( C , T ) , or is its isolated element, then the implicit difference equation has an asymptotic solution that is asymptotically constant, provided it has a bounded asymptotic solution. In the case of σ Γ ( C , T ) ⊂ { 1 } we prove that every asymptotic solution is asymptotically constant. Furthermore, we give an application of the result to periodic evolution equations associated with C-semigroups.

Analyzing Approximate and Exact Wave Solutions of Optical Fibres in the Time-fractional Estevez-Mansfield-Clarkson Equation

The study aims to employ the dynamics of hyperbolic and solitons wave solutions to the time-fractional Estevez-Mansfield Clarkson equation, which is utilize to stimulate the optical fibre waves in communication system and image processing. The proposed method entails transforming nonlinear fractional partial differential equations into nonlinear ordinary differential equations. Various solutions for the current model are derived in the form of exponential and trigonometric functions. The 2D and 3D under some suitable values of parameters are also plotted. The accomplished solutions show that these are reliable, applicable, efficient and intricate dynamics inherent in the communication system, image processing and might be used in further works to found novel solutions for other types of nonlinear evolution equations ascending in physical science and engineering. The novelty of our result is to offer new and critical insights into the intricate wave behaviors of the time-fractional Estevez-Mansfield-Clarkson equation by illustrating the basic mechanics of the importance of waves interaction and propagation in optical fibres in deepening our knowledge about the basic physical model.

A Study on Square-Mean S-Asymptotically Bloch Type Periodic Solutions for Some Stochastic Evolution Systems with Piecewise Constant Argument

This work is mainly focused on square-mean S-asymptotically Bloch type periodicity and its applications. The main aim of the paper is to introduce the definition of square-mean S-asymptotically Bloch type periodic processes with values in complex Hilbert spaces and systematically analyze some qualitative properties of this type of processes. These properties, combined with the inequality technique, evolution operator theory, fixed-point theory, and stochastic analysis approach, allow us to establish conditions for the existence and uniqueness of square-mean S-asymptotically Bloch type periodicity of bounded mild solutions for a class of stochastic evolution equations with infinite delay and piecewise constant argument. In the end, examples are given to illustrate the feasibility of our results.

A direct trial function approach for solving nonlinear evolution equations

The Klein equation, Infeld equation, and Sivashinsky equation not only start from realistic physical phenomena but can also be widely used in many physically significant fields such as plasma physics, fluid dynamics, crystal lattice theory, nonlinear circuit theory, and astrophysics. As a consequence, it is a very significant and challenging topic to research the explicit and accurate travelling wave solutions to these three equations. In this paper, in order to solve these three nonlinear partial differential equations (NPDEs), we have made some modifications to the trial function technique proposed by Xie and Tang by bringing in an ansatz solution containing two E-exponential functions. On this basis, we have developed a direct trial function technique to seek the explicit and accurate travelling wave solutions of nonlinear evolution equations (NEEs). We have illustrated its feasibility by applying it to the Klein equation, Infeld equation, and Sivashinsky equation. As a result, a lot of more general explicit and accurate travelling wave solutions of these three equations, including the solitary wave solutions and the singular travelling wave solutions, are successfully constructed in a straightforward and simple manner. The obtained solutions are quite equivalent to those given in the existing references. In addition, compared with the proposed approaches in the existing references, the approach described herein appears to be less calculative. Our technique may provide a novel way of thinking for solving NEEs. It is our firm conviction that the procedure used herein may also be utilized to explore the explicit and accurate travelling wave solutions of other NEEs. We try to generalize this approach to search for the explicit and accurate traveling wave solutions of other NEEs.

Geometric aspects of Raychaudhuri equation and implications in nonsingular universe

Raychaudhuri equation (RE) is a very important equation in gravity and cosmology as it is the key ingredient of the seminal singularity theorems. In this paper, we present a novel method of deriving the RE and restating the famous singularity theorems in GR using the matrix representation of tensor fields. For the first time in the literature, we introduce a Raychaudhuri matrix, whose evolution equation gives the matrix RE. Further, the paper deals with the consequences of RE in the context of nonsingular model of the universe like wormhole and emergent universe to highlight the fact that RE is not only important in hinting the singular nature of Einstein gravity but also bears importance in framing nonsingular models. The paper also identifies some transformations which can be used to restate the Focusing theorem. Further, it shows how RE is analogs to the differential equation of a real Harmonic oscillator and convergence condition is associated with the time varying frequency of the oscillator. Moreover, RE hints that exotic matter is needed for emergent model of the universe. Finally, the fate of time-like as well as null geodesics inside a wormhole and role of red-shift, shape function in convergence is analyzed from the point of view of RE.

A One-Step Modified New Iterative Method for Solving Partial Differential Equation

This study introduces a reliable semi-analytical approach for solving partial differential equations (PDEs) using a Modified New Iterative Method (MNIM). The primary aim is to enhance the efficiency of deriving closed-form solutions through an innovative formulation of an integral operator based on n-fold integration. This approach circumvents the conventional necessity of transforming PDEs into systems of multiple integral equations, thereby streamlining the solution process. The effectiveness of the MNIM is assessed through a series of examples, demonstrating its rapid convergence and superior performance in solving an array of evolution and partial differential equations. The results indicate that the MNIM not only simplifies the solution process but also significantly improves computational efficiency compared to traditional methods. This contribution holds substantial implications for both theoretical advancements in numerical analysis and practical applications across various fields where PDEs are prevalent, thereby facilitating more effective problem-solving strategies in complex systems.

Diverse wave solutions to the new extended (2 + 1)-dimensional nonlinear evolution equation: Phase portrait, bifurcation and sensitivity analysis, chaotic pattern, variational principle, and Hamiltonian

The purpose of this work is to give a deeper exploration into the nonlinear dynamics of the new extended [Formula: see text]-dimensional nonlinear evolution equation (NEE) for oceanic wave. Wielding the semi-inverse method (SIM) and traveling wave transformation, we establish the variational principle (VP). On basis of the VP, the corresponding system’s Hamiltonian is extracted. Aided by the Galilean transformation, the planar dynamical system is obtained. Then the phase portraits are plotted and the bifurcation analysis is presented to discuss the existing conditions of the wave solutions. In addition, the chaotic behaviors and sensitivity analysis of the system are also elaborated via adding the perturbed term and taking the different initial conditions, respectively. In the end, two robust methods, the variational method that stemmed from the VP and Ritz method, as well as the Hamiltonian-based method are used to develop the diverse wave solutions, which include the bell-shaped solitary, anti-bell-shaped solitary and periodic wave solutions. The findings of this study are all novel and can enable us to gain a deeper understanding of the nonlinear dynamics of the equation being studierd.