The (3 + 1)-dimensional Painlevé integrable equation are a class of nonlinear differential equations with special properties, which play an important role in nonlinear science and are of great ...significance in solving various practical problems, such as many important models in fields such as quantum mechanics, statistical physics, nonlinear optics, and celestial mechanics. In this work, we utilize the Hirota bilinear form and Mathematica software to formally obtain the interaction solution among lump wave, solitary wave and periodic wave, which has not yet appeared in other literature. Additionally, using the
-expansion method, we provide a rich set of exact solutions for the (3 + 1)-dimensional Painlevé integrable equation, which includes two functions with arbitrary values. This method is the first to be applied to the (3 + 1)-dimensional Painlevé integrable equation. By giving some 3D graphics and density maps, the dynamic properties are analyzed and demonstrated, which is beneficial for promoting understanding and application of the (3 + 1)-dimensional Painlevé integrable equation.
By utilizing the Hirota’s bilinear form and symbolic computation, abundant lump solutions and lump–kink solutions of the new (3 + 1)-dimensional generalized Kadomtsev–Petviashvili equation are ...derived in this work. Meanwhile, the interaction between lump solutions and the exponential function is also investigated. The dynamic properties of these obtained lump and interaction solutions are analyzed and described in figures by selecting appropriate parameters.
Fe/N/C is a promising non‐Pt electrocatalyst for the oxygen reduction reaction (ORR), but its catalytic activity is considerably inferior to that of Pt in acidic medium, the environment of polymer ...electrolyte membrane fuel cells (PEMFCs). An improved Fe/N/C catalyst (denoted as Fe/N/C‐SCN) derived from Fe(SCN)3, poly‐m‐phenylenediamine, and carbon black is presented. The advantage of using Fe(SCN)3 as iron source is that the obtained catalyst has a high level of S doping and high surface area, and thus exhibits excellent ORR activity (23 A g−1 at 0.80 V) in 0.1 M H2SO4 solution. When the Fe/N/C‐SCN was applied in a PEMFC as cathode catalyst, the maximal power density could exceed 1 W cm−2.
A non‐precious Fe/N/C electrocatalyst was prepared through pyrolysis of Fe(SCN)3, poly‐m‐phenylenediamine, and carbon black. The obtained Fe/N/C catalyst has high level of S doping and high surface area, and thus exhibits excellent catalytic activity for the oxygen reduction reaction in acidic solution. A polymer electrolyte membrane fuel cell using this catalyst as the cathode can yield a maximal power density as high as 1.03 W cm−2.
Based on a direct variable transformation, we obtain multiple rogue wave solutions of a generalized (3 + 1)-dimensional variable-coefficient nonlinear wave equation, including first-order, two-order ...and three-order rogue wave solutions. Their dynamic behaviors are shown by some 3D plots. Compared with Zha’s symbolic computation approach, we do not need to resort to Hirota bilinear form, and it can be used to deal with variable-coefficient integrable equations. Interaction solution between rogue wave and periodic wave is obtained by using the Hirota bilinear form. Abundant breather wave solutions are presented by a direct test function.
In this paper, a new (3+1)-dimensional Hirota bilinear equation in fluids is investigated. The interaction solutions of lump and
N
-soliton (
N
>
1
) are obtained. When
N
=
2
,
3
,
4
, new dynamical ...behavior is analyzed. The interaction of lump and periodic solutions is studied, including lump and 1-periodic solution and lump and 2-periodic solution. The interaction of lump, periodic and 1-soliton solutions is discussed. Finally, we obtain the multi-lump solutions, including 1-order, 2-order and 3-order lump solutions.
The adsorption performance of ZIF-67 towards 21 different organic dyes were investigated, and the corresponding mechanism was proposed.
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Zeolitic imidazolate frameworks-67 (ZIF-67) was ...prepared via electrochemical deposition, which was used to conduct adsorptive removal of various organic dyes. The results revealed that ZIF-67 displayed good adsorption performance towards some dyes due to its positive Zeta potential at wide pH range or some special functional groups like SO3 of dyes. Also, ZIF-67 could separate some dyes from the matrix subjected to its preferential uptake towards certain dyes. The possible adsorption mechanism including electrostatic interactions, coordination interactions and hydrogen bonding interactions was proposed and tested.
Most biochemical reactions in living cells are open systems interacting with environment through chemostats to exchange both energy and materials. At a mesoscopic scale, the number of each species in ...those biochemical reactions can be modeled by a random time-changed Poisson processes. To characterize macroscopic behaviors in the large number limit, the law of large numbers in the path space determines a mean-field limit nonlinear reaction rate equation describing the dynamics of the concentration of species, while the WKB expansion for the chemical master equation yields a Hamilton–Jacobi equation and the Legendre transform of the corresponding Hamiltonian gives the good rate function (action functional) in the large deviation principle. In this paper, we decompose a general macroscopic reaction rate equation into a conservative part and a dissipative part in terms of the stationary solution to the Hamilton–Jacobi equation. This stationary solution is used to determine the energy landscape and thermodynamics for general chemical reactions, which particularly maintains a positive entropy production rate at a non-equilibrium steady state. The associated energy dissipation law at both the mesoscopic and macroscopic levels is proved together with a passage from the mesoscopic to macroscopic one. A non-convex energy landscape emerges from the convex mesoscopic relative entropy functional in the large number limit, which picks up the non-equilibrium features. The existence of this stationary solution is ensured by the optimal control representation at an undetermined time horizon for the weak KAM solution to the stationary Hamilton–Jacobi equation. Furthermore, we use a symmetric Hamiltonian to study a class of non-equilibrium enzyme reactions, which leads to nonconvex energy landscape due to flux grouping degeneracy and reduces the conservative–dissipative decomposition to an Onsager-type strong gradient flow. This symmetric Hamiltonian implies that the transition paths between multiple steady states (rare events in biochemical reactions) is a modified time reversed least action path with associated path affinities and energy barriers. We illustrate this idea through a bistable catalysis reaction and compute the energy barrier for the transition path connecting two steady states via its energy landscape.
In this paper, a variable-coefficient symbolic computation approach is proposed to solve the multiple rogue wave solutions of nonlinear equation with variable coefficients. As an application, a (
2
+
...1
)-dimensional variable-coefficient Kadomtsev–Petviashvili equation is investigated. The multiple rogue wave solutions are obtained and their dynamic features are shown in some 3D and contour plots.
We propose a semi-discrete scheme for 2D Keller-Segel equations based on a symmetrization reformation, which is equivalent to the convex splitting method and is free of any nonlinear solver. We show ...that, this new scheme is stable as long as the initial condition does not exceed certain threshold, and it asymptotically preserves the quasi-static limit in the transient regime. Furthermore, we show that the fully discrete scheme is conservative and positivity preserving, which makes it ideal for simulations. The analogical schemes for the radial symmetric cases and the subcritical degenerate cases are also presented and analyzed. With extensive numerical tests, we verify the claimed properties of the methods and demonstrate their superiority in various challenging applications.
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