Showing papers on "Conservation law published in 2022"

Darboux transformation, localized waves and conservation laws for an M-coupled variable-coefficient nonlinear Schrödinger system in an inhomogeneous optical fiber

Abstract: Optical fiber communication system is one of the supporting systems in the modern internet age. We investigate an M-coupled variable-coefficient nonlinear Schrödinger system, which describes the simultaneous pulse propagation of the M-field components in an inhomogeneous optical fiber, where M is a positive integer. With respect to the complex amplitude of the jth-field (j=1,…,M) component in the optical fiber, we construct an n-fold Darboux transformation, where n is a positive integer. Based on the n-fold Darboux transformation, we obtain some one- and two-fold localized wave solutions for the above system with the mixed defocusing-focusing-type nonlinearity and M=2. We acquire the infinitely-many conservation laws. Via such solutions, we obtain some vector gray solitons, interactions between the two vector parabolic/cubic gray solitons, and interactions between the vector parabolic/cubic breathers and gray solitons with different β(z), γ(z) and δ(z), the coefficients of the group velocity dispersion, nonlinearity and amplification/absorption. It can be found that δ(z) affects the backgrounds of the breathers and gray solitons.

Predicting certain vector optical solitons via the conservation-law deep-learning method

Abstract: • The ECDL method integrates seamlessly data and mathematical physics models, even in partially understood, uncertain and high-dimensional contexts. • The ECDL method makes the predicted solution of physical model more consistent. • The ECDL method is effective for inverse problems and predicting the formation mechanism of vector solitons. The energy conservation law is introduced into a loss function of the physics-informed neural network (PINN), and an energy-conservation deep-learning (ECDL) method is constructed to study a coupled nonlinear Schrödinger equation (CNLSE). Using the ECDL method, we analyze the formation mechanism of vector solitons in birefringent fibers, and predict the dynamic behaviors of vector solitons, including one-soliton, two-soliton interaction, soliton molecule, rogue wave, and nondegenerate soliton. The related physical processes such as the energy conversion and power conservation along the propagation of soliton are studied. The results show that the nonlinear and dispersive effects separately cause the pulse broadening in time and frequency domains. The energy, shape and velocity of pulse in the transmission process remain unchanged when the two effects are balanced. Compared with the PINN method, the ECDL has higher accuracy and good generalization ability for a variety of soliton pulse propagation scenarios in optical fiber. Therefore, the deep learning method based on the prior knowledge of energy conservation is an effective tool to promote the research of nonlinear optics.

On the modified Gardner type equation and its time fractional form

Abstract: • Modified Gardner type equation and its time fractional form are derived. • From these two equations, we derived the nonlinear Schrodinger equation (NLS) type equation. • Symmetry analysis and conservation laws also presented. Differential equations play an important role in many scientific fields. In this work, we study modified Gardner-type equation and its time fractional form. We first derive these two equations from Fermi-Pasta-Ulam (FPU) model, and found that these two equations are related with nonlinear Schr o ¨ dinger equation (NLS) type of equations. Subsequently, symmetries and conservation laws are investigated. Finally, B a ¨ cklund transformation of conservation laws also presented. In this article, we not only derive these two equations, but also use perturbation analysis to find the connection between them and the Schr o ¨ dinger equation. Another key point is that B a ¨ cklund transformation of conservation laws are also obtained. From these results, it is obvious that the Lie group method is a very effective method for dealing with partial differential equations .

Nonlinear differential-difference hierarchy relevant to the Ablowitz-Ladik equation: Lax pair, conservation laws, N-fold Darboux transformation and explicit exact solutions

Abstract: Nonlinear differential-difference equations appear in optics, condensed matter physics, plasma physics and other fields. In this paper, we investigate a nonlinear differential-difference hierarchy relevant, in the case of θ=0, to the Ablowitz-Ladik equation, where θ=0,1. That hierarchy is obtained via a discrete spectral problem and the associated discrete spectral problem. When θ=1, Lax pair of the first nonlinear differential-difference system in that hierarchy is obtained. When θ=1, conservation laws and N-fold Darboux transformation of the first nonlinear differential-difference system in that hierarchy are derived with the aid of that Lax pair, where N is a positive integer. When θ=1, explicit exact solutions of that system are determined via that N-fold Darboux transformation. Discrete one soliton and interaction between the discrete one soliton and one breather-like wave are graphically depicted.

Physics-informed attention-based neural network for hyperbolic partial differential equations: application to the Buckley–Leverett problem

Abstract: Physics-informed neural networks (PINNs) have enabled significant improvements in modelling physical processes described by partial differential equations (PDEs) and are in principle capable of modeling a large variety of differential equations. PINNs are based on simple architectures, and learn the behavior of complex physical systems by optimizing the network parameters to minimize the residual of the underlying PDE. Current network architectures share some of the limitations of classical numerical discretization schemes when applied to non-linear differential equations in continuum mechanics. A paradigmatic example is the solution of hyperbolic conservation laws that develop highly localized nonlinear shock waves. Learning solutions of PDEs with dominant hyperbolic character is a challenge for current PINN approaches, which rely, like most grid-based numerical schemes, on adding artificial dissipation. Here, we address the fundamental question of which network architectures are best suited to learn the complex behavior of non-linear PDEs. We focus on network architecture rather than on residual regularization. Our new methodology, called physics-informed attention-based neural networks (PIANNs), is a combination of recurrent neural networks and attention mechanisms. The attention mechanism adapts the behavior of the deep neural network to the non-linear features of the solution, and break the current limitations of PINNs. We find that PIANNs effectively capture the shock front in a hyperbolic model problem, and are capable of providing high-quality solutions inside the convex hull of the training set.

Data-Driven Soliton Solutions and Model Parameters of Nonlinear Wave Models Via the Conservation-Law Constrained Neural Network Method

Abstract: In the process of the deep learning, we integrate more integrable information of nonlinear wave models, such as the conservation law obtained from the integrable theory, into the neural network structure, and propose a conservation-law constrained neural network method with the flexible learning rate to predict solutions and parameters of nonlinear wave models. As some examples, we study real and complex typical nonlinear wave models, including nonlinear Schrödinger equation, Korteweg-de Vries and modified Korteweg-de Vries equations. Compared with the traditional physics-informed neural network method, this new method can more accurately predict solutions and parameters of some specific nonlinear wave models even when less information is needed, for example, in the absence of the boundary conditions. This provides a reference to further study solutions of nonlinear wave models by combining the deep learning and the integrable theory.

A study of the generalized nonlinear advection-diffusion equation arising in engineering sciences

Abstract: In this work, we examine a nonlinear partial differential equation of fluid mechanics, namely, the generalized nonlinear advection–diffusion equation, which portrays the motion of buoyancy driven plume in a bent-on porous medium. Firstly, we classify all (point) symmetries of the equation, which prompt three cases of n. Next, for each case, we construct an optimal system of one-dimensional subalgebras and use them to perform symmetry reductions and symmetry invariant solutions. In a bid to explain the physical significance of some invariant solutions secured, we present a graphic display of some solutions in 3D, 2D as well as density plots via the exploitation of numerical simulations. Besides, we categorically state here that the results obtained in this study are new when compared with the outcomes previously achieved by Loubens et al., 2011 Quart. Appl. Math. 69 389–401. Interestingly, kink shape soliton, dark soliton, singular soliton together with exponential function solution wave profiles are displayed to make this work more valuable. Furthermore, we determine the conserved vectors in two different ways: engaging the general multiplier approach and Ibragimov’s conservation law theorem. Finally, we provide the physical meaning of these conservation laws.

Data-driven soliton solutions and model parameters of nonlinear wave models via the conservation-law constrained neural network method

Abstract: In the process of the deep learning, we integrate more integrable information of nonlinear wave models, such as the conservation law obtained from the integrable theory, into the neural network structure, and propose a conservation-law constrained neural network method with the flexible learning rate to predict solutions and parameters of nonlinear wave models. As some examples, we study real and complex typical nonlinear wave models, including nonlinear Schrödinger equation, Korteweg-de Vries and modified Korteweg-de Vries equations. Compared with the traditional physics-informed neural network method, this new method can more accurately predict solutions and parameters of some specific nonlinear wave models even when less information is needed, for example, in the absence of the boundary conditions. This provides a reference to further study solutions of nonlinear wave models by combining the deep learning and the integrable theory.

On fractional symmetry group scheme to the higher dimensional space and time fractional dissipative Burgers equation

Abstract: In this paper, we studied a higher-dimensional space and time fractional model, namely, the (3+1)-dimensional dissipative Burgers equation which can be used to describe the shallow water waves phenomena. Here, the analyzed tool is the Lie symmetry scheme in the sense of the Riemann–Liouville fractional derivative. First of all, the symmetry of this considered equation was yielded. Then, based on the above obtained symmetry, the one-parameter Lie group was obtained. Subsequently, this model can be changed into the lower-dimensional equation with the Erdélyi–Kober fractional operators. Lastly, conservation laws of this studied equation via a new conservation theorem were also received. After such a series of processing, these new results play an important role in our understanding of this higher-dimensional space and time differential equations.

Analytical and numerical study of chirped optical solitons in a spatially inhomogeneous polynomial law fiber with parity-time symmetry potential

Abstract: This work studies the dynamical transmission of chirped optical solitons in a spatially inhomogeneous nonlinear fiber with cubic-quintic-septic nonlinearity, weak nonlocal nonlinearity, self-frequency shift and parity-time (  ) symmetry potential. A generalized variable-coefficient nonlinear Schrödinger equation that models the dynamical evolution of solitons has been investigated by the analytical method of similarity transformation and the numerical mixed method of split-step Fourier method and Runge–Kutta method. The analytical self-similar bright and kink solitons, as well as their associated frequency chirps, are derived for the first time. We found that the amplitude of the bright and kink solitons can be controlled by adjusting the imaginary part of the  -symmetric potential. Moreover, the influence of the initial chirp parameter on the soliton pulse widths is quantitatively analyzed. It is worth emphasizing that we could control the chirp whether it is linear or nonlinear by adjusting optical fiber parameters. The simulation results of bright and kink solitons fit perfectly with the analytical ones, and the stabilities of these soliton solutions against noises are checked by numerical simulation.

First principles determination of bubble wall velocity

Abstract: The terminal wall velocity of a first-order phase transition bubble can be calculated from a set of fluid equations describing the scalar fields and the plasma's state. We rederive these equations from the energy-momentum tensor conservation and the Boltzmann equation, without linearizing in the background temperature and fluid velocity. The resulting equations have a finite solution for any wall velocity. We propose a spectral method to integrate the Boltzmann equation, which is simple, efficient and accurate. As an example, we apply this new methodology to the singlet scalar extension of the standard model. We find that all solutions are naturally categorized as deflagrations ($v_w\sim c_s$) or ultrarelativistic detonations ($\gamma_w\gtrsim10$). Furthermore, the contributions from out-of-equilibrium effects are, most of the time, subdominant. Finally, we use these results to propose several approximation schemes with increasing levels of complexity and accuracy. They can be used to considerably simplify the methodology while correctly describing the qualitative behavior of the bubble wall.

A study of the generalized nonlinear advection-diffusion equation arising in engineering sciences

Abstract: In this work, we examine a nonlinear partial differential equation of fluid mechanics, namely, the generalized nonlinear advection–diffusion equation, which portrays the motion of buoyancy driven plume in a bent-on porous medium. Firstly, we classify all (point) symmetries of the equation, which prompt three cases of n. Next, for each case, we construct an optimal system of one-dimensional subalgebras and use them to perform symmetry reductions and symmetry invariant solutions. In a bid to explain the physical significance of some invariant solutions secured, we present a graphic display of some solutions in 3D, 2D as well as density plots via the exploitation of numerical simulations. Besides, we categorically state here that the results obtained in this study are new when compared with the outcomes previously achieved by Loubens et al., 2011 Quart. Appl. Math. 69 389–401. Interestingly, kink shape soliton, dark soliton, singular soliton together with exponential function solution wave profiles are displayed to make this work more valuable. Furthermore, we determine the conserved vectors in two different ways: engaging the general multiplier approach and Ibragimov’s conservation law theorem. Finally, we provide the physical meaning of these conservation laws.

Triki–Biswas model: Its symmetry reduction, Nucci’s reduction and conservation laws

Abstract: In this paper, the symmetry reduction method and Nucci’s reduction method are used to obtain exact solutions to the Triki–Biswas equation. Furthermore, the new conservation theorem is utilized for finding the conservation laws of the given model. The conservation laws are derived for each admitted symmetry of the Triki–Biswas equation and satisfy the divergence condition. The 3D, contour and 2D figures are finally plotted to show the dynamics of the obtained exact solutions.

The Sharma–Tasso–Olver–Burgers equation: its conservation laws and kink solitons

Abstract: Abstract The present paper deals with the Sharma–Tasso–Olver–Burgers equation (STOBE) and its conservation laws and kink solitons. More precisely, the formal Lagrangian, Lie symmetries, and adjoint equations of the STOBE are firstly constructed to retrieve its conservation laws. Kink solitons of the STOBE are then extracted through adopting a series of newly well-designed approaches such as Kudryashov and exponential methods. Diverse graphs in 2 and 3D postures are formally portrayed to reveal the dynamical features of kink solitons. According to the authors’ knowledge, the outcomes of the current investigation are new and have been listed for the first time.

nn-PINNs: Non-Newtonian physics-informed neural networks for complex fluid modeling

Abstract: Time- and rate-dependent material functions in non-Newtonian fluids in response to different deformation fields pose a challenge in integrating different constitutive models into conventional computational fluid dynamic platforms. Considering their relevance in many industrial and natural settings alike, robust data-driven frameworks that enable accurate modeling of these complex fluids are of great interest. The main goal is to solve the coupled Partial Differential Equations (PDEs) consisting of the constitutive equations that relate the shear stress to the deformation and fully capture the behavior of the fluid under various flow protocols with different boundary conditions. In this work, we present non-Newtonian physics-informed neural networks (nn-PINNs) for solving systems of coupled PDEs adopted for complex fluid flow modeling. The proposed nn-PINN method is employed to solve the constitutive models in conjunction with conservation of mass and momentum by benefiting from Automatic Differentiation (AD) in neural networks, hence avoiding the mesh generation step. nn-PINNs are tested for a number of different complex fluids with different constitutive models and for several flow protocols. These include a range of Generalized Newtonian Fluid (GNF) empirical constitutive models, as well as some phenomenological models with memory effects and thixotropic timescales. nn-PINNs are found to obtain the correct solution of complex fluids in spatiotemporal domains with good accuracy compared to the ground truth solution. We also present applications of nn-PINNs for complex fluid modeling problems with unknown boundary conditions on the surface, and show that our approach can successfully recover the velocity and stress fields across the domain, including the boundaries, given some sparse velocity measurements.

Prediction of optical solitons using an improved physics-informed neural network method with the conservation law constraint

Abstract: In this work, based on the original physics-informed neural networks, we propose an improved physics-informed neural network method by combining the conservation laws. As one of the important integrable properties of nonlinear physical models, the conservation law can bring strong constraining force for the neural network to solve nonlinear physical models. Using this method, we study the standard nonlinear Schrödinger equation and predict various data-driven optical soliton solutions, including one-soliton, soliton molecules, two-soliton interaction, and rogue wave. In addition, from various exact solutions, we use the improved physics-informed neural network method to predict the dispersion and nonlinearity coefficients of the standard nonlinear Schrödinger equation based on the conservation law constraint. It turns out that the proposed method gives rise to the better results compared with the traditional physics-informed neural network method, and thus this method paves a way to simulate other physical models.

Highly dispersive optical solitons and conservation laws in absence of self–phase modulation with new Kudryashov's approach

Abstract: The new Kudryashov's approach obtains highly dispersive bright and singular optical solitons to the governing nonlinear Schrödinger's equation in absence of self–phase modulation. The derived bright soliton solution is applied to compute the conserved quantities from their densities obtained by the application of multipliers approach.

Optical Solitons and Conservation Laws for the Concatenation Model: Undetermined Coefficients and Multipliers Approach

Abstract: This paper retrieves an optical 1–soliton solution to a model that is written as a concatenation of the Lakshmanan–Porsezian–Daniel model and Sasa–Satsuma equation. The method of undetermined coefficients obtains a full spectrum of 1–soliton solutions. The multiplier approach yields the conserved densities, which subsequently lead to the conserved quantities from the bright 1–soliton solution.

On Thermodynamically Compatible Finite Volume Schemes for Continuum Mechanics

Abstract: In this paper we present a new family of semidiscrete and fully discrete finite volume schemes for overdetermined, hyperbolic, and thermodynamically compatible PDE systems. In the following, we will denote these methods as HTC schemes. In particular, we consider the Euler equations of compressible gasdynamics as well as the more complex Godunov--Peshkov--Romenski (GPR) model of continuum mechanics, which, with the aid of suitable relaxation source terms, is able to describe nonlinear elasto-plastic solids at large deformations as well as viscous fluids as two special cases of a more general first-order hyperbolic model of continuum mechanics. The main novelty of the schemes presented in this paper lies in the fact that we solve the entropy inequality as a primary evolution equation rather than the usual total energy conservation law. Instead, total energy conservation is achieved as a mere consequence of a thermodynamically compatible discretization of all the other equations. For this, we first construct a discrete framework for the compressible Euler equations that mimics the continuous framework of Godunov's seminal paper [Dokl. Akad. Nauk SSSR, 139(1961), pp. 521--523] exactly at the discrete level. All other terms in the governing equations of the more general GPR model, including nonconservative products, are judiciously discretized in order to achieve discrete thermodynamic compatibility, with the exact conservation of total energy density as a direct consequence of all the other equations. As a result, the HTC schemes proposed in this paper are provably marginally stable in the energy norm and satisfy a discrete entropy inequality by construction. We show some computational results obtained with HTC schemes in one and two space dimensions, considering both the fluid limit as well as the solid limit of the governing PDEs.