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...     

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 ( $$2+1$$...     

Interaction behaviors for the ( $$\varvec{2+1}$$ 2 + 1 )-dimensional Sawada–Kotera equation

Nonlinear Dynamics - The chimera state, exhibiting a hybrid state of coexisting coherent and incoherent behaviors, has become a fast growing field in the past decade. In this paper, we investigate...     

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.     

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...     

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.     

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...     

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...     

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...     

Existence and Stability of Time Periodic Solution to the Compressible Navier–Stokes–Korteweg System on $${\mathbb{R}^3}$$

The compressible Navier–Stokes–Korteweg system is considered on \({\mathbb{R}^3}\) when the external force is periodic in the time variable. The existence of a time periodic solution is proved for a sufficiently small external force by using the time-T-map related to the linearized problem around the motionless state with constant density and absolute temperature. The spectral properties of the time-T-map is investigated by a potential theoretic method and an energy method in some weighted spaces. The stability of the time periodic solution is proved for sufficiently small initial perturbations. It is also shown that the \({L^\infty}\) norm of the perturbation decays as time goes to infinity.     

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).

     

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...     

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.     

A direct Bäcklund transformation for a (3+1)-dimensional Kadomtsev–Petviashvili–Boussinesq-like equation

In this paper, a Kadomtsev–Petviashvili–Boussinesq-like equation in (3+1)-dimensions is firstly introduced by using the combination of the Hirota bilinear Kadomtsev–Petviashvili equation and Boussinesq equation in terms of function f. And then a direct bilinear Bäcklund transformation of this new model is constructed, which consists of seven bilinear equations and ten arbitrary parameters. Based on this constructed bilinear Bäcklund transformation, some classes of exponential and rational traveling wave solutions with arbitrary wave numbers are presented.     

Solutions and conservation laws of a (3+1)-dimensional Zakharov–Kuznetsov equation

In this paper, we study a nonlinear evolution partial differential equation, namely the (3+1)-dimensional Zakharov–Kuznetsov equation. Kudryashov method together with Jacobi elliptic function method is used to obtain the exact solutions of the (3+1)-dimensional Zakharov–Kuznetsov equation. Furthermore, the conservation laws of the (3+1)-dimensional Zakharov–Kuznetsov equation are obtained by using the multiplier method.     

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.     

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.     

Lie symmetry analysis and invariant solutions of $$\varvec{}$$-dimensional Calogero–Bogoyavlenskii–Schiff equation

Lie group analysis is applied to carry out the similarity reductions of the \((3+1)\)-dimensional Calogero–Bogoyavlenskii–Schiff (CBS) equation. We obtain generators of infinitesimal transformations of the CBS equation and each of these generators depend on various parameters which give us a set of Lie algebras. For each of these Lie algebras, Lie symmetry method reduces the \((3+1)\)-dimensional CBS equation into a new \((2+1)\)-dimensional partial differential equation and to an ordinary differential equation. In addition, we obtain commutator table of Lie brackets and symmetry groups for the CBS equation. Finally, we obtain closed-form solutions of the CBS equation by using the invariance property of Lie group transformations.     

Non-linear electromechanical response of 1–3 type piezocomposites

Domain switching in piezoelectric materials is caused by external loads such as electric field and stress that leads to non-linear behaviour. A study is carried out to compare the non-linear behaviour of 1–3 piezocomposites with different volume fractions and bulk piezoceramics. Experiments are conducted to measure the electrical displacement and strain on piezocomposites and bulk ceramics under high cyclic electrical loading and constant compressive prestress. A thermodynamically consistent uni-axial framework is developed to predict the nonlinear behaviour by combining the phenomenological and micromechanical techniques. Volume fractions of three distinct uni-axial variants (instead of six variants) are used as internal variables to describe the microscopic state of the material. In this model, the grain boundary effects are taken into account by introducing the back fields (electric field and stress) as non-linear kinematic hardening functions. An analytical model based on equivalent layered approach is used to calculate effective properties such as elastic, piezoelectric, and dielectric constants for different volume fractions of piezocomposites. The predicted effective properties are incorporated in the proposed uni-axial model and the dielectric hysteresis (electrical displacement versus electric field) as well as butterfly curves (strain versus electric field) are simulated. Comparison between the experiments and simulations show that this model can reproduce the characteristics of non-linear response. It is observed that the variation in fiber volume fraction and compressive stress has a significant influence on the response of the 1–3 piezocomposites.     

One-dimensional optical solitons in cubic–quintic–septimal media with $$\varvec{\mathcal {}}$$-symmetric potentials

A (\(1+1\))-dimensional inhomogeneous cubic–quintic–septimal nonlinear Schrödinger equation with \(\mathcal {PT}\)-symmetric potentials is studied, and two families of soliton solutions are obtained. From soliton solutions, the amplitude of soliton is independent of the \(\mathcal {PT}\)-symmetric potential parameter k; however, the phase depends on the parameter k. The phase of soliton alters from negative to positive values at the location of center. Moreover, the evolutional behaviors of these solitons are discussed.