Abundant fission and fusion solutions in the ( $$2+1$$ 2 + 1 )-dimensional generalized Calogero–Bogoyavlenskii–Schiff equation

Nonlinear Dynamics - Fission and fusion are important phenomena, which have been observed experimentally in many physical areas. In this paper, we study the above two phenomena in the ( $$2+1$$...     

New mixed solutions generated by velocity resonance in the $$(2+1)$$ ( 2 + 1 ) -dimensional Sawada–Kotera equation

Nonlinear Dynamics - Based on the N-soliton solutions of the $$(2+1)$$ -dimensional Sawada–Kotera equation, the collisions among lump waves, line waves, and breather waves are studied in this...     

Exact solutions with elastic interactions for the (2 $$+$$ + 1)-dimensional extended Kadomtsev–Petviashvili equation

Nonlinear Dynamics - In this work, the $$(2+1)$$ -dimensional extended Kadomtsev–Petviashvili equation, which models the surface waves and internal waves in straits or channels, is...     

Studies on the breather solutions for the $$\mathbf{(2+1)}$$ -dimensional variable-coefficient Kadomtsev–Petviashvili equation in fluids and plasmas

Nonlinear Dynamics - In this paper, we study the $$(2 + 1)$$ -dimensional variable-coefficient Kadomtsev–Petviashvili equation, which has certain applications in fluids and plasmas. Via the...     

Bright soliton interactions in a $$\mathbf (2 +\mathbf 1) $$ ( 2 + 1 ) -dimensional fourth-order variable-coefficient nonlinear Schrödinger equation for the Heisenberg ferromagnetic spin chain

Nonlinear Dynamics - Interactions of bright solitons in the Heisenberg ferromagnetic spin chain, governed by a $$(2+1)$$ -dimensional nonlinear Schrödinger equation with variable coefficients,...     

Nonlinear interaction of solitary waves described by multi-rational wave solutions of the (2 $$+$$ + 1)-dimensional Kadomtsev–Petviashvili equation with variable coefficients

Nonlinear Dynamics - In this paper, the generalized unified method is used to construct multi-rational wave solutions of the ( $$2 + 1$$ )-dimensional Kadomtsev–Petviashvili equation with...     

Solving the $$\mathbf{(3+1) }$$ ( 3 + 1 ) -dimensional KP–Boussinesq and BKP–Boussinesq equations by the simplified Hirota’s method

Nonlinear Dynamics - We study two (3 $$+$$ 1)-dimensional generalized equations, namely the Kadomtsev–Petviashvili–Boussinesq equation and the B-type...     

Bäcklund transformation,rogue wave solutions and interaction phenomena for a $$\varvec{(3+1)}$$ ( 3 + 1 ) -dimensional B-type Kadomtsev–Petviashvili–Boussinesq equation

Nonlinear Dynamics - Under investigation in this paper is the $$(3+1)$$ -dimensional B-type Kadomtsev–Petviashvili–Boussinesq (BKP–Boussinesq) equation, which can display the...     

The $$\varvec{N}$$ N -soliton,fusion, rational and breather solutions of two extensions of the (2+1)-dimensional Bogoyavlenskii–Schieff equation

Nonlinear Dynamics - The aim of this work is to analyze and explore the dynamics of two extensions of the Bogoyavlenskii–Schieff equation. The Hirota bilinear method is applied to the...     

Interaction solutions for a reduced extended $$\mathbf{(3}\varvec{+}{} \mathbf{1)}$$ ( 3 + 1 ) -dimensional Jimbo–Miwa equation

Heavy-duty industrial robots have great advantages in the manufacturing industry. Considering the heavy process load and low stiffness of the robot, an accurate and efficient dynamic model plays an important role in the behavior analysis and performance improvement in the robot. This paper presents a novel methodology for the inverse dynamic analysis of the heavy-duty industrial robot with elastic joints. In particular, high-order kinematics and dynamics are concisely deduced using Lie group to deal with elastic joints for the robot inverse dynamic analysis. Meanwhile, position errors of the end-effector due to elastic joints are evaluated through the inverse dynamic analysis when the robot is in heavy-duty applications. Compared with previous approaches, the advantage of proposed method is that new formulas for inverse dynamic analysis are shown to be more concise and computationally efficient using Lie group. Moreover, the position error evaluation method considering dynamic forces is proved to be more accurate than the traditional method when the robot is in the high-speed application. Because of the high computational efficiency and accurate evaluation results, the proposed approach is applicable to trajectory optimization and position error compensation, especially for the robot in heavy-load and high-speed applications.     

Evolution behaviour of kink breathers and lump- $$\pmb {M}$$ M -solitons ( $$\pmb {M\rightarrow \infty }$$ M → ∞ ) for the (3+1)-dimensional Hirota–Satsuma–Ito-like equation

Nonlinear Dynamics - In this paper, some novel lump solutions and interaction phenomenon between lump and kink M-soliton are investigated. Firstly, we study the evolution and degeneration behaviour...     

Lump-type solution and breather lump–kink interaction phenomena to a ($$\mathbf{{3{\varvec{+}}1}}$$3+1)-dimensional GBK equation based on trilinear form

Nonlinear Dynamics - In this paper, the multivariate trilinear operators in the ($$3+1$$)-dimensional space are applied to a ($$3+1$$)-dimensional GBK equation. The resulting trilinear form is used...     

Symmetry of Traveling Wave Solutions to the Allen–Cahn Equation in $${{\mathbb R^{2}}}$$

In this paper, we prove even symmetry of monotone traveling wave solutions to the balanced Allen–Cahn equation in the entire plane. Related results for the unbalanced Allen–Cahn equation are also discussed.     

Transfer model and kinetic characteristics of $${\rm NH}_{4}^{+}$$ –K+ ion exchange on K-zeolite

Based on the mass transfer theory, a new mass transfer model of ion-exchange process on zeolite under liquid film diffusion control is established, and the kinetic curves and the mass transfer coefficients of –K⁺ ion-exchange under different conditions were systemically determined using the shallow-bed experimental method. The results showed that the –K⁺ ion-exchange rates and transfer coefficients are directly proportional to solution flow rate and temperature, and inversely proportional to solution viscosity and the size of zeolite granules. It also showed that the transfer coefficient is not influenced by the ion concentrations. For a large ranges of operational conditions including temperatures (10 − 75°C), flow rates (0.031 m s⁻¹ −0.26 m s⁻¹), liquid viscosities (1.002 × 10⁻³ N s m⁻² − 4.44 × 10⁻³ N s m⁻²), and zeolite granular sizes (0.2 − 1.45 mm), the average mass transfer coefficients calculated by the model agree with the experimental results very well.     

SBV Regularity for Hamilton–Jacobi Equations in $${{\mathbb R}^n}$$

In this paper we study the regularity of viscosity solutions to the following Hamilton–Jacobi equations

$\partial_{t}u+H(D_{x}u)=0\quad\hbox{in }\Omega\subset{\mathbb R}\times{\mathbb R}^{n}.$

In particular, under the assumption that the Hamiltonian \({H\in C^2({\mathbb R}^n)}\) is uniformly convex, we prove that D _x u and ? _t u belong to the class SBV _loc (Ω).

     

A k–$${\varepsilon}$$ turbulence closure model of an isothermal dry granular dense matter

The turbulent flow characteristics of an isothermal dry granular dense matter with incompressible grains are investigated by the proposed first-order k–\({\varepsilon}\) turbulence closure model. Reynolds-filter process is applied to obtain the balance equations of the mean fields with two kinematic equations describing the time evolutions of the turbulent kinetic energy and dissipation. The first and second laws of thermodynamics are used to derive the equilibrium closure relations satisfying turbulence realizability conditions, with the dynamic responses postulated by a quasi-linear theory. The established closure model is applied to analyses of a gravity-driven stationary flow down an inclined moving plane. While the mean velocity decreases monotonically from its value on the moving plane toward the free surface, the mean porosity increases exponentially; the turbulent kinetic energy and dissipation evolve, respectively, from their minimum and maximum values on the plane toward their maximum and minimum values on the free surface. The evaluated mean velocity and porosity correspond to the experimental outcomes, while the turbulent dissipation distribution demonstrates a similarity to that of Newtonian fluids in turbulent shear flows. When compared to the zero-order model, the turbulent eddy evolution tends to enhance the transfer of the turbulent kinetic energy and plane shearing across the flow layer, resulting in more intensive turbulent fluctuation in the upper part of the flow. Solid boundary as energy source and sink of the turbulent kinetic energy becomes more apparent in the established first-order model.     

Optimal Time Decay of the Vlasov–Poisson–Boltzmann System in $${\mathbb R^3}$$

The Vlasov–Poisson–Boltzmann System governs the time evolution of the distribution function for dilute charged particles in the presence of a self-consistent electric potential force through the Poisson equation. In this paper, we are concerned with the rate of convergence of solutions to equilibrium for this system over \mathbb R³{\mathbb R^3}. It is shown that the electric field, which is indeed responsible for the lowest-order part in the energy space, reduces the speed of convergence, hence the dispersion of this system over the full space is slower than that of the Boltzmann equation without forces; the exact L ²-rate for the former is (1 + t)^−1/4 while it is (1 + t)^−3/4 for the latter. For the proof, in the linearized case with a given non-homogeneous source, Fourier analysis is employed to obtain time-decay properties of the solution operator. In the nonlinear case, the combination of the linearized results and the nonlinear energy estimates with the help of the proper Lyapunov-type inequalities leads to the optimal time-decay rate of perturbed solutions under some conditions on initial data.     

二次非线性粘弹性圆板的2/1+3/1超谐解

计及材料的非线性弹性和粘性性质，研究了圆板在简谐载荷作用下的2／1＋3／1超谐解，导出了相应的非线性动力方程。提出一类强非线性动力系统的叠加－叠代谐波平衡法。将描述动力系统的二阶常微分方程，化为基本解为未知函数的基本微分方程；及分岔解为未知函数的增量微分方程。通过叠加－迭代谐波平衡法得出了圆板的2／1＋3／1超谐解。对叠加迭代谐波平衡法和数值积分法进行了比较，两者结果吻合很高。并且讨论了2／1＋3／1超谐解的渐近稳定性。     

High-order solutions of motion near triangular libration points for arbitrary value of $${\varvec{\mu }}$$ μ

Nonlinear Dynamics - In this paper, a new methodology is proposed to derive the high-order approximations of motions near them in three cases, i.e., the mass ratio is greater than, smaller than and...     

Nonlinear Instability in Gravitational Euler–Poisson Systems for $$\gamma=\frac{6}{5}$$

The dynamics of gaseous stars can be described by the Euler–Poisson system. Inspired by Rein’s stability result for , we prove the nonlinear instability of steady states for the adiabatic exponent under spherically symmetric and isentropic motion.