Conservation Laws Research Articles

Viscoelastic plastic creep constitutive model based on energy conservation law and strain energy theory

Viscoelastic plastic creep constitutive model based on energy conservation law and strain energy theory

A Moment-Based Hermite WENO Scheme with Unified Stencils for Hyperbolic Conservation Laws

A Moment-Based Hermite WENO Scheme with Unified Stencils for Hyperbolic Conservation Laws

Stability of quasi-entropy solutions of non-local scalar conservation laws

Stability of quasi-entropy solutions of non-local scalar conservation laws

Euler–Lagrange equation for gradient-type Lagrangian and related conservation laws

AbstractVariational calculus with gradient-type variations has often been neglected, although it proves to be suitable for certain concrete problems governed by several evolution variables. These kinds of variations lead to Euler–Lagrange partial differential equations controlled by the right-hand member. In this context, we also introduce anti-trace Euler–Lagrange partial differential equations that are suitable for some innovative ideas. Also, some applications are provided for the theoretical results derived in the paper.

Electro-hyperelasticity with phase field for electric breakdown modeling

In this paper, we systematically derive the phase field model for electric breakdown channel propagation. The model includes electric charge, mass, momentum and energy conservation equations as well as Allen–Cahn type equation which governs the phase field evolution. The medium motion is described by the Godunov–Romenski type hyperelastic model. Electromagnetic part of the model consists of Maxwell’s equations in quasi(electro)stationary approximation including electric charge conservation law. Derivation of the model is based on the Coleman–Noll procedure and Gurtin’s microforces and microstress framework.

Self-organization of collisionless shocks: from a laminar profile to a rippled time-dependent structure

A collisionless shock structure results from the nonlinear interaction between charged particles and electromagnetic fields. Yet, a collisionless shock is globally governed by the mass, momentum and energy conservation requirements. A stable shock structure must ensure that the fluxes of the conserved quantities are constant on average, and, therefore, is determined by this necessity. Here, we study an observed high upstream temperature high Mach number shock and show that the conservation laws cannot be fulfilled unless the shock is spatially inhomogeneous along the shock front and time-dependent.

Lie symmetries, exact solutions and conservation laws of time fractional coupled (2+1)-dimensional nonlinear Schrödinger equations

ABSTRACT In this paper, Lie symmetry analysis method is applied to time fractional coupled (2 + 1)-dimensional nonlinear Schrödinger equations. We obtain all the Lie symmetries and reduce the (2 + 1)-dimensional fractional partial differential equations with Riemann-Liouville fractional derivative to (1 + 1)-dimensional counterparts with Erdélyi-Kober fractional derivative. Then we obtain the power series solutions and prove their convergence. In addition, the conservation laws for the governing equations are constructed by using the generalization of the Noether operators and the new conservation theorem.

Lie symmetries, exact solution and conservation laws of (2 + 1)-dimensional time fractional Kadomtsev–Petviashvili system

Abstract In this paper, Lie symmetry analysis method is applied to the ( 2 + 1 ) (2+1) -dimensional time fractional Kadomtsev–Petviashvili (KP) system, which is an important model in mathematical physics. We obtain all the Lie symmetries admitted by the KP system and use them to reduce the ( 2 + 1 ) (2+1) -dimensional fractional partial differential equations with Riemann–Liouville fractional derivative to some ( 1 + 1 ) (1+1) -dimensional fractional partial differential equations with Erdélyi–Kober fractional derivative or Riemann–Liouville fractional derivative, thereby getting some exact solutions of the reduced equations. In addition, the new conservation theorem and the generalization of Noether operators are developed to construct the conservation laws for the system studied.

Dynamic plastic damage analysis of bending-torsion coupling in cross stiffener systems under mass impact

ABSTRACT This study proposes a simpler theoretical model to predict the bending-torsion coupling dynamic plastic response of cross stiffener systems under mass impact. Traditional methods often rely on torque balance and momentum conservation equations, which involve complex and difficult calculations. To simplify this process, the study uses the law of energy conservation and applies the variational method to solve the dynamic plastic damage of the cross stiffener system. The plastic response of a single stiffener is first analyzed and compared with existing theories, showing consistent results for maximum displacement. The study then extends the model to more complex conditions, performing comparative analyses on stiffeners of different sizes through numerical simulation. The findings show that the maximum displacement at the load and node remains within a reasonable range, with a displacement difference of only 0.9 mm and an error margin of 4.79% in one of the test conditions.

Non‐self‐similar solutions containing delta shock waves for nonhomogeneous zero‐pressure gas dynamics with energy conservation law by the viscosity method

This article considers the zero‐pressure gas dynamics governed by the conservation laws of mass, momentum, and energy with time‐dependent source terms. By adopting the vanishing viscosity method, we propose a viscous system to prove the stability of non‐self‐similar Riemann solutions including the delta shock wave and vacuum.

The Two-Thirds Power Law Derived from a Higher-Derivative Action

The two-thirds power law is a link between angular speed ω and curvature κ observed in voluntary human movements: ω is proportional to κ2/3. Squared jerk is known to be a Lagrangian leading to the latter law. However, it leads to unbounded movements and is therefore incompatible with quasi-periodic dynamics, such as the movement of the tip of a pen drawing ellipses. To solve this drawback, we give a class of higher-derivative Lagrangians that allow for both quasi-periodic and unbounded movements, and at the same time lead to the two-thirds power law. The current study extends this framework and investigates a wider class of Lagrangians admitting generalised conservation laws.

Nonlinear waves for a variable-coefficient modified Kadomtsev–Petviashvili system in plasma physics and electrodynamics

In this paper, a variable-coefficient modified Kadomtsev–Petviashvili (vcmKP) system is investigated by modeling the propagation of electromagnetic waves in an isotropic charge-free infinite ferromagnetic thin film and nonlinear waves in plasma physics and electrodynamics. Painlevé analysis is given out, and an auto-Bäcklund transformation is constructed via the truncated Painlevé expansion. Based on the auto-Bäcklund transformation, analytic solutions are given, including the solitonic, periodic and rational solutions. Using the Lie symmetry approach, infinitesimal generators and symmetry groups are presented. With the Lagrangian, the vcmKP equation is shown to be nonlinearly self-adjoint. Moreover, conservation laws for the vcmKP equation are derived by means of a general conservation theorem. Besides, the physical characteristics of the influences of the coefficient parameters on the solutions are discussed graphically. Those solutions have comprehensive implications for the propagation of solitary waves in nonuniform backgrounds.

Conservation laws for a perturbed resonant nonlinear Schrödinger equation in quantum fluid dynamics and quantum optics

The current paper retrieves the conservation laws for the extended version of the resonant nonlinear Schrödinger's equation for description of physical processes in quantum fluid dynamics and quantum optics. The method of multipliers recovers three fundamental conservation laws. Analytical solutions of equation are found taking into account traveling wave reduction. The conserved quantities are subsequently computed from the soliton solution of the model equation that is derived in this work too.

Computational Technology for Shell Models of Magnetohydrodynamic Turbulence Constructing

The paper discusses the computational technology for constructing one type of small-scale magnetohydrodynamic turbulence models – shell models. Any such model is a system of ordinary quadratic nonlinear differential equations with constant coefficients. Each phase variable is interpreted in absolute value as a measure of the intensity of one of the fields of the turbulent system in a certain range of spatial scales (scale shell). The equations of any shell model must have several quadratic invariants, which are analogues of conservation laws in ideal magnetohydrodynamics. The derivation of the model equations consists in obtaining such expressions for constant coefficients for which the predetermined quadratic expressions will indeed be invariants. Derivation of these expressions «manually» is quite cumbersome and the likelihood of errors in formula transformations is high. This is especially true for non-local models in which large-scale shells that are distant in size can interact. The novelty and originality of the work lie in the fact that the authors proposed a computational technology that allows one to automate the process of deriving equations for shell models. The technology was implemented using computer algebra methods, which made it possible to obtain parametric classes of models in which the invariance of given quadratic forms is carried out absolutely accurately – in formula form. The determination of the parameter values in the resulting parametric class of models is further carried out by agreement with the measures of the interaction of shells in the model with the probabilities of their interaction in a real physical system. The idea of the described technology and its implementation belong to the authors. Some of its elements were published by the authors earlier, but in this work, for the first time, its systematic description is given for models with complex phase variables and agreement of measures of interaction of shells with probabilities. There have been no similar works by other authors previously. The technology allows you to quickly and accurately generate equations for new non-local turbulence shell models and can be useful to researchers involved in modeling turbulent systems.

Superposition of soliton, breather and lump waves in a non-painlevé integrabale extension of the Boiti-Leon-Manna-Pempinelli equation

Abstract In this paper, we derive general $N$th-order Pfaffian solutions for a $(3+1)$-dimensional non-Painlev{'e} integrable extension of the Boiti-Leon-Manna-Pempinelli (BLMP) equation. Specifcally, we obtain $N$-soliton, higher-order breather, higher-order lump and hybrid solutions, and explore the superpositions of Y-shaped and X-shaped soliton-breather waves.
Moreover, we construct bilinear B"{a}cklund transformations, Lax pairs, and conservation laws using Bell polynomials.
Finally, we identify a similar equation in the literature and demonstrate that it represents another non-Painlevé integrable extension of the BLMP equation.

Conservation laws, traveling wave solutions and wavelet solution for the two-component Novikov equation

Abstract The objective of this present article is to study the two-component Novikov equation. This
 equation is also known as Geng-Xue equation and is characterized by cubic nonlinearities. By
 applying the multiplier method, new conserved quantities are found, and some traveling wave
 solutions are derived based on the (G′/G )-expansion method. Furthermore, a numerical solution
 is determined by employing the Haar wavelet collocation method. In order to confirm the
 accuracy, this numerical solution is compared with the solution obtained by (G′/G )-expansion
 method. Additionally, the convergence analysis of the Haar wavelet collocation method is also
 presented.

Reflection parity and space-time parity photonic conservation laws in parametric nonlinear optics

Conservation laws are some of the most generic and useful concepts in physics. In nonlinear optical parametric processes, conservation of photonic energy, momenta and parity often lead to selection rules, restricting the allowed polarization and frequencies of the emitted radiation. Here we present a scheme to derive conservation laws in optical parametric processes in which many photons are annihilated and a single photon is emitted. We first rederive with it the known nonlinear optical conservation laws, and then utilize it to predict and explore conservations of reflection parity and space-time parity. Conservation of arises from a generalized reflection symmetry of the polarization in a superspace, analogous to the superspace employed in the study of quasicrystals. Conservation of similarly arises from space-time reversal symmetry in superspace. We explore these conservation laws numerically in the context of high-harmonic generation and outline experimental setups where they can be tested. Published by the American Physical Society 2024

Analisis Pemahaman Konsep Hukum Kekekalan Energi Mekanik Mahasiswa Tadris IPA Pada Mata Kuliah Energi Dalam Sistem Kehidupan

Physics is a branch of science that contains complex explanations regarding the relationships between various natural events which are expressed as facts, theories, concepts, principles and physical laws. Energy is one of essential chapter in physics which contains concepts that need to be understood not memorized. One of the topics discussed in energy material which includes many abstract concepts is the law of conservation of mechanical energy. Studying the law of conservation of mechanical energy, which contains abstract concepts, certainly requires conceptual understanding not just remembering and memorizing. Understanding concepts in physics is very important because it can help develop critical thinking skill, creative thinking skill, and good problem solving skill. The descriptive quantitative method with a test approach was used as a method in this study and the sample is 72 Tadris IPA college students of IAIN Ponorogo in the 3rd semester. The results of the study show that in the category of understanding the concept of the law of conservation of energy in objects moving vertically upwards was classified as moderate, in the category understanding the concept of the law of conservation of energy in objects moving along a certain path was classified as low.

Upwind block-centered multistep differences for semiconductor device problem with heat and magnetic influences

In this paper, a mathematical model of semiconductor device with two important factors is discussed. Heat and magnetic influences are considered together. Its numerical approximation is important in information science and other technological innovation fields. The major physical unknowns (the potential, electron concentration, hole concentration, and the heat) are defined by four nonlinear PDEs: an elliptic equation, two convection-diffusion equations and a heat conductor equation. A conservative block-centered method is used to approximate the elliptic equation. The derivatives to time t, ∂∂t(⋅), are discretized by multistep differences for improving the computational accuracy. The diffusions and convection terms are treated by block-centered differences and upwind approximations, respectively. This composite numerical method is suitable for this nonlinear system. Firstly, numerical dispersion and nonphysical oscillations are avoided. Secondly, the unknowns and their adjoint vectors are obtained simultaneously. Thirdly, the law of conservation is preserved. An error estimates is derived. Finally, simple examples are given for showing the efficiency and accuracy.

Locally linearized physics-informed neural networks for Riemann problems of hyperbolic conservation laws

In this work, we demonstrate that physics-informed neural networks (PINNs) tend to propagate predicted shock wave information bidirectionally in time, which does not align with the actual evolution direction of solutions to hyperbolic conservation laws. This mismatch results in instability and hinders the reduction of the loss of governing equations, as well as the initial condition loss by the deep neural network. In order to tackle this problem, we simplify the complexity of the problem by constructing equivalent linear transport equations in the region of shock wave generation. The speeds of these linearized waves are governed by the Rankine–Hugoniot relations of conservation laws. This approach is termed the Locally Linearized PINNs method. Specifically, an appropriate shock wave detector is initially designed to identify domains where shock waves occur. Near shock waves, the original nonlinear equations are transformed into their linearized forms, thereby modifying the residual terms of the partial differential equations. Additionally, an equilibrium factor is introduced in fluid compression regions to reduce prediction errors and stabilize the training of deep neural networks. Numerical examples illustrate that Locally Linearized PINNs effectively address the challenge of predicting global solutions in PINNs and significantly improve shock-capturing performance for hyperbolic conservation laws.