Nonclassical Symmetries Research Articles

Resonant multi-wave, positive multi-complexiton, nonclassical Lie symmetries, and conservation laws to a generalized Hirota bilinear equation

This paper explores the evolutionary behavior of some specific waves for a Generalized Hirota Bilinear (GHB) equation with applications in modern sciences. More precisely, the linear superposition principle is first adopted to construct the resonant multi-wave of the GHB equation. Through the resonant N-wave and some particular computations, positive multi-complexiton to the governing model is then extracted with the use of computational packages. Furthermore, after deriving nonclassical Lie symmetries and their corresponding invariant solutions, Conservation Laws (CLs) for the GHB equation are formally constructed based on a general method developed by Ibragimov. The propagation dynamics of resonant double- and triple-waves as well as positive single- and double-complexitons are examined in detail by considering several case studies. The present results demonstrate how adjusting the nonlinear parameter can be used to control specific waves in nonlinear systems.

Classical and nonclassical Lie symmetries, bifurcation analysis, and Jacobi elliptic function solutions to a 3D-modified nonlinear wave equation in liquid involving gas bubbles

The current paper undertakes an in-depth exploration of the dynamics of nonlinear waves governed by a 3D-modified nonlinear wave equation, a significant model in the study of complex wave phenomena. To this end, the study employs both classical and nonclassical Lie symmetries for rigorously deriving invariant solutions of the governing equation. These symmetries enable the formal construction of exact solutions, which are crucial for understanding the complex behavior of the model. Furthermore, the research extends into the realm of bifurcation analysis through the application of planar dynamical system theory. Such an analysis reveals the conditions under which the 3D-modified nonlinear wave equation admits Jacobi elliptic function solutions. The study also delves into the impact of the nonlinear parameter on the physical characteristics of bright and kink solitary waves as well as continuous periodic waves using Maple. Overall, the comprehensive analysis presented not only enhances the understanding of complex nonlinear wave dynamics but also sets the stage for future advancements in vast areas of fluid dynamics and plasma physics.

Symmetries and Separation of Variables

In this paper, we look at the method of separation of variables of a PDE from its symmetry transformation point of view. Specifically, we discuss the relation between the existence of additively and multiplicatively separated variables of a PDE, and the form of its symmetry operators. We show that solutions in the form of separated variables are in fact, invariant solutions, i.e. solutions invariant under some subalgebra of the symmetry operators of the equation. For the case of two independent variables, we obtain the form of Lie point symmetry operators corresponding to additively and multiplicatively separated solutions, and generalize our results for the case when separated variables are any functions of independent variables. We also discuss the role of contact symmetry transformations and differential invariants for the existence of separated solutions, and outline the role of variational symmetries, as well as conditional (non-classical) symmetry operators. We demonstrate that the symmetry approach is a valuable tool for obtaining information regarding existence of solutions with separated variables.

Analytic shock‐fronted solutions to a reaction–diffusion equation with negative diffusivity

AbstractReaction–diffusion equations (RDEs) model the spatiotemporal evolution of a density field according to diffusion and net local changes. Usually, the diffusivity is positive for all values of , which causes the density to disperse. However, RDEs with partially negative diffusivity can model aggregation, which is the preferred behavior in some circumstances. In this paper, we consider a nonlinear RDE with quadratic diffusivity that is negative for . We use a nonclassical symmetry to construct analytic receding time‐dependent, colliding wave, and receding traveling wave solutions. These solutions are multivalued, and we convert them to single‐valued solutions by inserting a shock. We examine properties of these analytic solutions including their Stefan‐like boundary condition, and perform a phase plane analysis. We also investigate the spectral stability of the and constant solutions, and prove for certain and that receding traveling waves are spectrally stable. In addition, we introduce a new shock condition where the diffusivity and flux are continuous across the shock. For diffusivity symmetric about the midpoint of its zeros, this condition recovers the well‐known equal‐area rule, but for nonsymmetric diffusivity it results in a different shock position.

Nonclassical symmetries, optimal classification, and dynamical behavior of similarity solutions of (3+1)-dimensional Burgers equation

This paper is devoted to the study of symmetry group classification, and optimal subalgebras of the three-dimensional Burgers equation, which describes nonlinear wave motion exhibits nonlinear effects and enhanced dispersion in the context of nonlinear dynamics. The equation undergoes a Painlevé analysis to assess its integrability properties to verify the possibility that it passes the Painlevé test, followed by the derivation of nonclassical symmetries. Utilizing the invariance property of Lie groups, desirable infinitesimal symmetries of Lie algebra have been outlined for the equation. Relying on the invariance characteristic of adjoint transformation along with the newly generated similarity variables, an intensive and systematically organized approach is employed to achieve a one-dimensional optimal system. The entire set of group invariant solutions for each of the associated subalgebras has been established. The dynamical behavior of derived exact solutions is examined using numerical simulations, and numerous intriguing occurrences are discovered, such as flat sheet, periodic wave, doubly soliton, multisoliton, bright-dark soliton, breather soliton, line soliton type, which offer valuable insights into the dynamics of the system and can predict the behavior of waves in real-world scenarios.

Non-Lie Reduction Operators and Potential Transformations for a Special System with Applications in Plasma Physics

Non-Lie reduction operators, also known as nonclassical symmetries, are derived for special systems that appear in Plasma Physics. These operators are used to construct similarity mappings, which reduce the systems under study into systems of ordinary differential equations. Furthermore, potential equivalence transformations are presented. Based on these results, a number of exact solutions are constructed.

Non-Classical Symmetry Analysis of a Class of Nonlinear Lattice Equations

In this paper, a non-classical symmetry method for obtaining the symmetries of differential–difference equations is proposed. The non-classical symmetry method introduces an additional constraint known as the invariant surface condition, which is applied after the infinitesimal transformation. By solving the governing equations that satisfy this condition, we can obtain the corresponding reduced equation. This allows us to determine the non-classical symmetry of the differential–difference equation. This method avoids the complicated calculation involved in extending the infinitesimal generator and allows for a wider range of symmetry forms. As a result, it enables the derivation of a greater number of differential–difference equations. In this paper, two kinds of (2+1)-dimensional Toda-like lattice equations are taken as examples, and their corresponding symmetric and reduced equations are obtained using the non-classical symmetry method.

Classical and non-classical symmetries of time-fractional Navier–Stokes equation

Classical and non-classical symmetries of time-fractional Navier–Stokes equation

The $$(2 + 1)$$-dimensional generalized time-fractional Zakharov Kuznetsov Benjamin Bona Mahony equation: its classical and nonclassical symmetries, exact solutions, and conservation laws

The $$(2 + 1)$$-dimensional generalized time-fractional Zakharov Kuznetsov Benjamin Bona Mahony equation: its classical and nonclassical symmetries, exact solutions, and conservation laws

Exact solutions of hyperbolic reaction-diffusion equations in two dimensions

Exact solutions are constructed for a class of nonlinear hyperbolic reaction-diffusion equations in two-space dimensions. Reduction of variables and subsequent solutions follow from a special nonclassical symmetry that uncovers a conditionally integrable system, equivalent to the linear Helmholtz equation. The hyperbolicity is commonly associated with a speed limit due to a delay, \(\tau\), between gradients and fluxes. With lethal boundary conditions on a circular domain wherein a species population exhibits logistic growth of Fisher–KPP type with equal time lag, the critical domain size for avoidance of extinction does not depend on \(\tau\). A diminishing exact solution within a circular domain is also constructed, when the reaction represents a weak Allee effect of Huxley type. For a combustion reaction of Arrhenius type, the only known exact solution that is finite but unbounded is extended to allow for a positive \(\tau\). doi: 10.1017/S1446181123000093

Symmetries and Exact Solutions of the Diffusive Holling–Tanner Prey-Predator Model

We consider the classical Holling–Tanner model extended on 1D space by introducing the diffusion term. Making a reasonable simplification, the diffusive Holling–Tanner system is studied by means of symmetry based methods. Lie and Q-conditional (nonclassical) symmetries are identified. The symmetries obtained are applied for finding a wide range of exact solutions, their properties are studied and a possible biological interpretation is proposed. 3D plots of the most interesting solutions are drown as well.

Nonclassical Symmetries, Nonlinear Self-adjointness, Conservation Laws and Some New Exact Solutions of Cylindrical KdV Equation

Nonclassical Symmetries, Nonlinear Self-adjointness, Conservation Laws and Some New Exact Solutions of Cylindrical KdV Equation

Invariant Solutions and Conservation Laws of the Time-Fractional Telegraph Equation

In this study, the Lie symmetry analysis is given for the time-fractional telegraph equation with the Riemann–Liouville derivative. This equation is useable to describe the physical processes of models possessing memory. By applying classical and nonclassical Lie symmetry analysis for the telegraph equation with α , β time-fractional derivatives and some technical computations, new infinitesimal generators are obtained. The actual methods give some classical symmetries while the nonclassical approach will bring back other symmetries to these equations. The similarity reduction and conservation laws to the fractional telegraph equation are found.

Analysis of Brownian motion in stochastic Schrödinger wave equation using Sardar sub-equation method

In this paper, the stochastic nonlinear Schrödinger equation (NLSE) forced by multiplicative noise that represents the temporal change of fluctuations is examined in a more extended form. Symmetry reduction (nonclassical symmetry) is used to convert the nonlinear partial differential equation into a nonlinear ordinary differential equation to obtain new optical soliton solutions. The Sardar sub-equation method is suggested to solve this stochastic nonlinear problem. The obtained stochastic solitons illustrate the dispersion of waves in optical fiber transmissions. This approach enables us to study a wide range of solutions with significant physical perspectives, including dark solitons, bright solitons, singular and periodic solitons with their noise term effects involved in Brownian motion based on the Itô sense. The noise term effects on the solitons are presented using 3D and 2D graphs. The results and calculations show the significance, accuracy and efficiency of the method. Various stable and unstable nonlinear stochastic differential equations that are found in mathematics, physics and other applied areas can be solved using this technique.

Enforcing continuous symmetries in physics-informed neural network for solving forward and inverse problems of partial differential equations

As a typical application of deep learning, physics-informed neural network (PINN) has been successfully used to find numerical solutions of partial differential equations (PDEs), but how to improve the limited accuracy is still a great challenge for PINN. In this work, we introduce a new method, symmetry-enhanced physics informed neural network (SPINN) where the invariant surface conditions induced by the Lie symmetries or non-classical symmetries of PDEs are embedded into the loss function in PINN, to improve the accuracy of PINN for solving the forward and inverse problems of PDEs. We test the effectiveness of SPINN for the forward problem via two groups of ten independent numerical experiments using different numbers of collocation points and neurons per layer for the Korteweg-de Vries (KdV) equation, breaking soliton equation, heat equation, and potential Burgers equations respectively, and for the inverse problem by considering different layers and neurons as well as different numbers of training points with different levels of noise for the Burgers equation in potential form. The numerical results show that SPINN performs better than PINN with fewer training points and simpler architecture of neural network, and in particular, exhibits superiorities than the PINN method and the two-stage PINN method of Lin and Chen by considering the Sawada-Kotera equation. Furthermore, we discuss the computational overhead of SPINN in terms of the relative computational cost to PINN and show that the training time of SPINN has no obvious increases, even less than PINN for certain cases.

Conditionally integrable PDEs, non-classical symmetries and applications

Some multi-dimensional nonlinear partial differential equations reduce to a linear equation in fewer dimensions after imposing one additional constraint. A large class of useful conditionally integrable reaction–diffusion equations follows from a single non-classical symmetry reduction, yielding an infinite-dimensional but incomplete solution space. This solution device extends further to other nonlinear equations such as higher-order reaction–diffusion, fractional-order diffusion and diffusion–convection. Applications are shown for population dynamics with or without weak Allee effects, speed-limited hyperbolic diffusion, material phase field dynamics, soil–water–plant dynamics and calcium transport on the curved surface of an oocyte. Beyond non-classical symmetry analysis, examples of other conditionally integrable equations are given; nonlinear diffusion in 1 + 2 dimensions and the Madelung–Burgers quantum fluid in 1 + 3 dimensions.

Nonclassical symmetry analysis and heir-equations of forced Burger equation with time variable coefficients

Nonclassical symmetry analysis and heir-equations of forced Burger equation with time variable coefficients

The Shigesada–Kawasaki–Teramoto model: Conditional symmetries, exact solutions and their properties

We study a simplification of the well-known Shigesada–Kawasaki–Teramoto model, which consists of two nonlinear reaction–diffusion equations with cross-diffusion. A complete set of Q-conditional (nonclassical) symmetries is derived using an algorithm adopted for the construction of conditional symmetries. The symmetries obtained are applied for finding a wide range of exact solutions, possible biological interpretation of some of which being presented. Moreover, an alternative application of the simplified model related to the polymerization process is suggested and exact solutions are found in this case as well.

The new stochastic solutions for three models of non-linear Schrödinger’s equations in optical fiber communications via Itô sense

In this paper, we consider three models of non-linear Schrödinger’s equations (NLSEs) via It\^{o} sense. Specifically, we study these equations forced by multiplicative noise via the Brownian motion process. There are numerous approaches for converting non-linear partial differential equations (NPDEs) into ordinary differential equations (ODEs) to extract wave solutions. The majority of these methods are a type of symmetry reduction known as non-classical symmetry. We apply the unified technique based on symmetry reduction to produce some new optical soliton solutions for the proposed equations. The obtained stochastic solutions depict the propagation of waves in optical fiber communications. The theoretical analysis and proposed results clarify that the presented technique is sturdy, appropriate, and efficacious. Some graphs of selected solutions are also depicted with the help of the MATLAB packet program. Indeed, the structure, bandwidth, amplitude, and phase shift are controlled by the influences of physical parameters in the presence of noise term via It\^{o} sense. Our results show that the proposed technique is better suited for solving many other complex models arising in real-life problems.

Classical and non-classical Lie symmetry analysis, conservation laws and exact solutions of the time-fractional Chen–Lee–Liu equation

Classical and non-classical Lie symmetry analysis, conservation laws and exact solutions of the time-fractional Chen–Lee–Liu equation