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We consider large-scale dynamics of non-equilibrium dense soliton gas for the Korteweg–de Vries (KdV) equation in the special “condensate” limit. We prove that in this limit the integro-differential kinetic equation for the spectral density of states reduces to the N -phase KdV–Whitham modulation equations derived by Flaschka et al. (Commun Pure Appl Math 33(6):739–784, 1980) and Lax and Levermore (Commun Pure Appl Math 36(5):571–593, 1983). We consider Riemann problems for soliton condensates and construct explicit solutions of the kinetic equation describing generalized rarefaction and dispersive shock waves. We then present numerical results for “diluted” soliton condensates exhibiting rich incoherent behaviors associated with integrable turbulence.

Boundary conditions required for numerical solution of the Boltzmann kinetic equation (BKE) for mass/heat transfer between evaporation and condensation surfaces are analyzed by comparison of BKE results with molecular dynamics (MD) simulations. Lennard–Jones potential with parameters corresponding to solid argon is used to simulate evaporation from the hot side, nonequilibrium vapor flow with a Knudsen number of about 0.02, and condensation on the cold side of the condensed phase. The equilibrium density of vapor obtained in MD simulation of phase coexistence is used in BKE calculations for consistency of BKE results with MD data. The collision cross-section is also adjusted to provide a thermal flux in vapor identical to that in MD. Our MD simulations of evaporation toward a nonreflective absorbing boundary show that the velocity distribution function (VDF) of evaporated atoms has the nearly semi-Maxwellian shape because the binding energy of atoms evaporated from the interphase layer between bulk phase and vapor is much smaller than the cohesive energy in the condensed phase. Indeed, the calculated temperature and density profiles within the interphase layer indicate that the averaged kinetic energy of atoms remains near-constant with decreasing density almost until the interphase edge. Using consistent BKE and MD methods, the profiles of gas density, mass velocity, and temperatures together with VDFs in a gap of many mean free paths between the evaporation and condensation surfaces are obtained and compared. We demonstrate that the best fit of BKE results with MD simulations can be achieved with the evaporation and condensation coefficients both close to unity.

In this paper, a new class of multivariable special functions and their generalizations is introduced and used to solve generalized fractional differential and kinetic equations. By applying the Sumudu transform, we derive solutions for the fractional differential equations and fractional kinetic equations expressed in terms of Prabhakar’s Mittag–Leffler function and Wiman’s Mittag–Leffler function, respectively. In contrast to previous research, which mostly focused on single‐variable Mittag–Leffler formulations, our method demonstrates the advantage of dealing with multivariable parameters, providing more versatility for modeling complicated fractional systems. With the use of illustrative examples that demonstrate application, this work is innovative in that it unifies and generalizes a number of previously proven results into a unified analytical framework. These results give fractional models, which may find use in physics, engineering, and other applied sciences, a better mathematical basis. MSC2020 Classification 26A33, 33E12, 44A10, 44A05, 44A35

Macroscopic dynamics of soliton gases can be analytically described by the thermodynamic limit of the Whitham equations, yielding an integro-differential kinetic equation for the density of states. Under a delta-functional ansatz, the kinetic equation for soliton gas reduces to a non-diagonalisable system of hydrodynamic type whose matrix consists of several 2 × 2 Jordan blocks. Here we demonstrate the integrability of this system by showing that it possesses a hierarchy of commuting hydrodynamic flows and can be solved by an extension of the generalised hodograph method. Our approach is a generalisation of Tsarev’s theory of diagonalisable systems of hydrodynamic type to quasilinear systems with non-trivial Jordan block structure.

In this paper, we introduce a new extension of the incomplete second Appell hypergeometric matrix functions (EISAHMFs) and extension of the second Appell hypergeometric matrix functions (ESAHMFs) in terms of the extended incomplete Pochhammer matrix symbols and extended Pochhammer matrix symbols, respectively. Firstly, we define an extension of the EISAHMFs and ESAHMFs. Subsequently, we investigate integral representations, differential formulas, transformation formulas, and recurrence relations pertaining to EISAHMFs. Then, we present the Mellin matrix transform of the ESAHMFs. Finally, a new result regarding the solution of the fractional kinetic equations in the terms of newly ESAHMFs is discussed.

The aim of the present paper is to develop a new generalized form of the fractional kinetic equation involving a generalized k-Mittag-Leffler function Ek,ζ,ηγ,ρ(⋅). The solutions of fractional kinetic equations are discussed in terms of the Mittag-Leffler function. Further, numerical values of the results and their graphical interpretation is interpreted to study the behavior of these solutions. The results established here are quite general in nature and capable of yielding both known and new results.

We investigate Hamiltonian aspects of the integro-differential kinetic equation for dense soliton gas which results as a thermodynamic limit of the Whitham equations. Under a delta-functional ansatz, the kinetic equation reduces to a non-diagonalisable system of hydrodynamic type whose matrix consists of several 2 × 2 Jordan blocks. We demonstrate that the resulting system possesses local Hamiltonian structures of differential-geometric type, for all standard two-soliton interaction kernels (KdV, sinh-Gordon, hard-rod, Lieb–Liniger, DNLS, and separable cases). In the hard-rod case, we show that the continuum limit of these structures provides a local multi-Hamiltonian formulation of the full kinetic equation.

Phosphate sorption on rendzina soil was studied by P-32 heterogeneous isotopic exchange under a steady-state. There are two types of sorbed phosphate, namely strongly and weakly bonded phosphate, the latter being able to exchange with phosphate (H 2 32 PO 4 − ) ions in the soil solution. The experimental kinetic data was not fitted by the one exponential kinetic model. Starting from this observation, a new kinetic model is established by assuming two types of weakly bonded phosphate, which take part in the two parallel exchange processes. A biexponential kinetic equation is obtained, which fits the experimental data much better than the one exponential equation. Graphical abstract

In this paper, we consider the long-term behavior of some special solutions to the Wave Kinetic Equation. This equation provides a mesoscopic description of wave systems interacting nonlinearly via the cubic NLS equation. Escobedo and Velázquez showed that, starting with initial data given by countably many Dirac masses, solutions remain a linear combination of countably many Dirac masses at all times. Moreover, there is convergence to a single Dirac mass at long times. The first goal of this paper is to give quantitative rates for the speed of said convergence. In order to study the optimality of the bounds we obtain, we introduce and analyze a toy model accounting only for the leading order quadratic interactions.

The properties and microstructure evolution of quaternary Cu-Ni-Co-Si alloys with different Ni/Co mass ratios were investigated. The microstructure and morphological characteristics of the precipitates were analyzed by using electron backscatter diffraction (EBSD), transmission electron microscopy (TEM) and high-resolution transmission electron microscopy (HRTEM). The mechanical properties and conductivity of the alloys were significantly improved after the addition of Co. The grains presented an obvious growth trend with an increase in Ni/Co mass ratios, and the appropriate Co accelerated the recrystallization process. The δ-(Ni, Co)2Si phases of the Cu-Ni-Co-Si alloys and δ-Ni2Si phases of the Cu-Ni-Si alloys shared the same crystal structure and orientation relationships with the matrix, which had two variant forms: δ1 and δ2 phases. The precipitates preferential grew along with the direction of the lowest energy and eventually exhibited two different morphologies. Compared with that of the Cu-Ni-Si alloy, the volume fraction of precipitates in the alloys with Co was significantly improved, accompanied by an increase in the precipitated phase size. The addition of Co promoted the precipitation of the precipitated phase and further purified the matrix. A theoretical calculation was conducted for different strengthening mechanisms, and precipitation strengthening was the key reinforcement mechanism. Moreover, the kinetic equations of both alloys were obtained and coincided well with the experimental results.