Group Invariant Solutions Research Articles

Some aspects of the group invariant solutions of wave propagation for an electric field

Some aspects of the group invariant solutions of wave propagation for an electric field

Fundamental solutions and conservation laws for conformable time fractional partial differential equation

In this paper, the connections between fundamental solutions and Lie symmetry groups for a class of conformable time fractional partial differential equations (PDEs) with variable coefficient are investigated for the first time. The group-invariant solutions to the considered equations are constructed by means of symmetry group method. Then, the corresponding fundamental solutions for these PDEs are established by taking the inverse Laplace transform of the group invariant solutions. In addition, some examples are introduced to illustrate the effectiveness of this approach. Furthermore, the conservation laws of these fractional PDEs are obtained making use of new Noether theorem.

Similarity Reductions, Power Series Solutions, and Conservation Laws of the Time-Fractional Mikhailov–Novikov–Wang System

The current study presents a comprehensive Lie symmetry analysis for the time-fractional Mikhailov–Novikov–Wang (MNW) system with the Riemann–Liouville fractional derivative. The corresponding simplified equations with the Erdélyi–Kober fractional derivative are constructed by group invariant solutions. Furthermore, we obtain explicit solutions with the help of the power series method and show the dynamical behavior via evolutional figures. Finally, by means of Ibragimov’s new conservation theorem, the conservation laws are derived for the system.

Symmetry analysis and closed-form invariant solutions of the nonlinear wave equations in elasticity using optimal system of Lie subalgebra

Nonlinear wave equations are generated by the elastic wave propagation through inelastic material. We studied such a unidirectional nonlinear elastic wave by considering potential having geometrical nonlinearity. The nonlinear problem in elasticity is studied by symmetry method. Optimal system of non-similar one dimensional subalgebras of Lie algebra is constructed. Symmetry reductions are performed using these subalgebras and corresponding group invariant solutions are found. The problem of nonlinear elastic wave with a damping term is also studied. Multiplier approach is used for finding conservation laws of these equations.

Optimal classification and similarity solution for unsteady flow behind a shock wave in a perfectly conducting dusty gas with magnetic field using group invariance method

We have studied the one dimensional unsteady isothermal and adiabatic flows behind the shock waves in a non-ideal dusty gas under the effect of magnetic field using Lie group invariance method. This method is used to get the infinitesimal generators of the Lie algebra. The concept of optimal systems is used for minimizing the group-invariant solutions. The optimal system of subalgebra is used to obtain the similarity variables and similarity transformation which transform the set of partial differential equations (PDEs) into a set of ordinary differential equations (ODEs). Consequently, the system of ODEs in different cases is solved by using Runge–Kutta (R-K) method of forth order to derive the similarity solution in the cases of isothermal and adiabatic flows. The detailed discussions are illustrated by graph through numerical calculation in the case of power-law shock path. It is shown that the shock wave strength has decaying effects with an increase in the values of Alfven Mach number, adiabatic exponent of the gas, and gas non-idealness parameter. Also, the shock strength increases with an increment in the values of mass concentration of solid particles, magnetic field variation exponent, the ratio of density of solid particles to the gas initial density, and by changing the geometry from cylindrical to spherical.

Novel symmetric structures and explicit solutions to a coupled Hunter-Saxton equation

In the current study, novel symmetric structures to a coupled Hunter-Saxton equation are synthetically investigated. These novel symmetric structures include Lie symmetries, discrete symmetries, nonlocally related systems, and μ-symmetries. Lie symmetries and μ-symmetries are then used to derive explicit invariant solutions. Based on the established optimal system, the coupled Hunter-Saxton equation can be reduced to rich ordinary differential equations by the Lie group transformation. Its group invariant solutions are thus obtained. Discrete symmetries to the coupled Hunter-Saxton equation are constructed utilizing Lie symmetries, which can help calculate new solutions from known explicit solutions. Moreover, nonlocally related systems of the coupled Hunter-Saxton equation are completed, which contain potential systems and inverse potential systems based on conservation laws and Lie symmetries, respectively. Furthermore, without using the group theory, more plentiful similarity reductions and similarity solutions to the coupled Hunter-Saxton equation are produced by employing the direct reduction method. Another class of symmetric structures to the coupled Hunter-Saxton equation explored in this paper are μ-symmetries, which are given by matching an integrable and horizontal one-form μ = Λ x dx + Λ t dt for Lie symmetries. Hence, μ-reductions, explicit solutions and μ-conservation laws can be determined by μ-symmetries. In addition, polynomial solutions are researched by considering the linear invariant subspaces admitted by the coupled Hunter-Saxton equation. Several explicit invariant solutions are described by graphs ultimately.

Symmetry analysis of the constant acceleration curve equation

Abstract Lie group theory is applied to the curve equation which maintains constant normal accelerations for a vehicle with constant deceleration. The curve equation is a third order nonlinear ordinary differential equation for which the symmetries are calculated. It is shown that the equation possesses four-parameter Lie group of transformations including scaling, rotation and translational symmetries. In the case of constant velocity, the algebra increases to a six-parameter Lie group of transformations. Using the symmetries of the differential equation, the group invariant solutions are determined first. The conditions for group invariant solutions to exist are given. By employment of the symmetries, a solution is obtained by reduction of order also. It is found that the nontrivial solutions are of implicit complex forms.

Lie group analysis of the general Karmarkar condition

The Karmarkar embedding condition in different spherically symmetrical metrics is studied in general using Lie symmetries. In this study, the Lie symmetries for conformally flat and shear-free metrics are studied which extend recent results. The Lie symmetries for geodesic metrics and general spherical spacetimes are also obtained for the first time. In all cases group invariant exact solutions to the Karmarkar embedding condition are obtained via a Lie group analysis. It is further demonstrated that the Karmarkar condition can be used to produce a model with interesting features: an embeddable relativistic radiating star with a barotropic equation of state via Lie symmetries.

Some novel physical structures of a (2+1)-dimensional variable-coefficient Korteweg–de Vries system

In this paper, we study the novel nonlinear wave structures of a (2+1)-dimensional variable-coefficient Korteweg–de Vries (KdV) system by its analytic solutions. Its N-soliton solutions are obtained via Hirota’s bilinear method, and in particular, the hybrid solutions of lump, breather and line solitons are derived by the long wave limit method. In addition to soliton solutions, similarity reduction, including similarity solutions (also known as group-invariant solutions) and novel non-autonomous rational third-order Painlevé equations, is achieved through symmetry analysis. The analytic results, together with illustrative wave interactions, show interesting physical features, that may shed some light on the study of other variable-coefficient nonlinear systems.

Group invariant solutions of wave propagation in phononic materials based on the reduced micromorphic model via optimal system of Lie subalgebra

In this paper the system of equations arising from a simplified micromorphic model is studied using the Lie symmetry approach. The advantage of this approach is that is provides exact invariant solutions rather than numerical or approximate ones reported in the literature in earlier studies. We obtain the Lie point symmetries for the model and the one dimensional optimal system of Lie subalgebras. Using the one-dimensional optimal system, symmetry reductions are performed and some corresponding invariant solutions are found. To illustrate the physical importance of group invariant solutions, 2D, 3D and contour plots are drawn.

Dual-mode nonlinear Schrödinger equation (DMNLSE): Lie group analysis, group invariant solutions, and conservation laws

This paper proposed a novel dual-mode nonlinear Schrödinger equation (DMNLSE) with cubic law. The DMNLSE describes the propagations of two moving waves simultaneously. This nonlinear Schrödinger equation (NLSE) model appears in different areas of physics, including nonlinear optics, plasma physics, superconductivity, and quantum mechanics. In the context of photonics, NLSE models the propagation of soliton pulses through inter-continental distances. In this study, the Lie symmetry procedure is successfully utilized on the DMNLSE to discover the infinitesimal generators using the invariance condition. Then, we transform the DMNLSE into an ordinary differential equation (ODE) employing accepted generators and similarity reduction concepts. Later, we consider conservation laws for the DMNLSE. Thanks to the multiplier technique, the conserved quantities’ densities and fluxes are discovered. With the aid of the Mathematica package program, 3D, contour, and density graphs were drawn for the special values of the parameters in the solutions, and the physical structures of the solutions obtained in this way were also observed.

An optimal System of Lie Subalgebras and Group-Invariant Solutions with Conserved Currents of a (3+1)-D Fifth-Order Nonlinear Model I with Applications in Electrical Electronics, Chemical Engineering and Pharmacy

AbstractHigher-dimensional nonlinear integrable partial differential equations are significant as they often describe diverse phenomena in nonlinear systems like laser radiations in a plasma, optical pulses in the glass fibres, fluid mechanics, radio waves in the ion sphere, condensed matter and electromagnetics. This article shows an analytical investigation of a (3+1)-dimensional fifth-order nonlinear model with KdV forming its main part. Lie group analysis of the model is performed through which its infinitesimal generators are obtained. These generators are engaged in the construction of an optimal system of Lie subalgebra in one dimension. Moreover, members of the system secured are utilized in reducing the underlying model to ordinary differential equations (ODEs) for possible exact solutions. In consequence, we achieve various functions, ranging from trigonometric, logarithmic, rational, to hyperbolic. In addition, the results found constitute diverse solitonic solutions such as complex, topological kink and anti-kink, trigonometric and bright. We utilize the power series technique to obtain a series solution of the most complicated ordinary differential equation with forty-four terms. In addition, we reveal the dynamics of these solutions via graphical depictions. In the end, we constructed conserved currents of the underlying equation through the use of the multiplier technique. Further, we utilize the optimal system of the underlying model to derive more conserved vectors using Ibragimov’s theorem for conservation laws.

GROUP ANALYSIS FOR KLEIN-GORDON EQUATION VIA THEIR SYMMETRIES

In the present paper we have obtained the one parameter groups and symmetry transformations associated to the classical symmetries of the Klein-Gordon (KG) equation, we have also constructed an optimal system of two dimensional sub-algebras of the KG equation which provides the preliminary classification of group invariant solutions and yield the most general group invariant solution.

Lie symmetry analysis, conservation laws and diverse solutions of a new extended (2+1)-dimensional Ito equation

<abstract><p>In this paper, a new class of extended (2+1)-dimensional Ito equations is investigated for its group invariant solutions. The Lie symmetry method is employed to transform the nonlinear Ito equation into an ordinary differential equation. The general solution of the solvable linear differential equation with different parameters is obtained, and the plot of the solvable linear differential equation is given. A power series solution for the equation is then derived. Furthermore, a conservation law for the equation is constructed by utilizing a new Ibragimov conservation theorem.</p></abstract>

Lie symmetry group, exact solutions and conservation laws for multi-term time fractional differential equations

<abstract><p>In this paper, the time fractional Benjamin-Bona-Mahony-Peregrine (BBMP) equation and time-fractional Novikov equation with the Riemann-Liouville derivative are investigated through the use of Lie symmetry analysis and the new Noether's theorem. Then, we construct their group-invariant solutions by means of Lie symmetry reduction. In addition, the power-series solutions are also obtained with the help of the Erdélyi-Kober (E-K) fractional differential operator. Furthermore, the conservation laws for the time-fractional BBMP equation are established by utilizing the new Noether's theorem.</p></abstract>

Existence and multiplicity results for a singular fourth-order elliptic system involving critical homogeneous nonlinearities

<abstract><p>This paper deals with a singular fourth-order elliptic system involving critical homogeneous nonlinearities. The existence and multiplicity results of group invariant solutions are established by variational methods and the Hardy-Rellich inequality.</p></abstract>

A new (3+1) dimensional Hirota bilinear equation: Painlavé integrability, Lie symmetry analysis, and conservation laws

This study's subject is a (3 + 1) dimensional new Hirota bilinear (NHB) equation that appears in the theory of shallow water waves. We investigate how particular dispersive waves behave in an NHB equation. In this regard, we first test the Painlavé integrability using the WTC-Kruskal method and second perform Lie point symmetry analysis on NHB. The algorithm outputs the Lie algebra, the symmetry reductions, and group invariant solutions. Using Lie point symmetries, NHB transforms into an ordinary differential equation. The integration architectures to solve this equation are the Bernoulli sub-ODE, 1/ G, and modified Kudryashov methods. We plot their graphs and observe how the solutions behaved in understanding the physical phenomenon. Additionally, we discuss the model utilizing nonlinear self-adjointness and Ibragimov's approach to generate conservation laws for each Lie symmetry generator.

Multiple Exp-Function Solutions, Group Invariant Solutions and Conservation Laws of a Generalized (2+1)-dimensional Hirota-Satsuma-Ito Equation

Multiple exp-function technique and group analysis is accomplished for a comprehensive (2+1)-dimensional Hirota-Satsuma-Ito equation that appears in many sectors of nonlinear science such as for example in fluid dynamics. Travelling wave solutions are computed and it is displayed that this underlying equation gives kink solutions. The invariant reductions and further closed-form solutions are processed. Conserved currents are developed and their physical ramifications are illustrated.

Symmetries and symmetry reductions of the combined KP3 and KP4 equation

To find symmetries, symmetry groups and group invariant solutions are fundamental and significant in nonlinear physics. In this paper, the finite point symmetry group of the combined KP3 and KP4 (CKP34) equation is found by means of a direct method. The related point symmetries can be obtained simply by taking the infinitesimal form of the finite point symmetry group. The point symmetries of the CKP34 equation constitute an infinite dimensional Kac-Moody–Virasoro algebra. The point symmetry invariant solutions of the CKP34 equation are obtained via the standard classical Lie point symmetry method.

Abundant invariant and classical solutions with the conservation laws of a new (3+1)-dimensional fifth-order nonlinear Wazwaz equation with the third-order dispersion terms in ocean physics

Ocean plays key roles in the earth’s system in shaping the planet’s climate as well as weather by absorbing, storing, and also transporting huge quantities of carbon-dioxide, water, heat and moisture. This paper presents the analytical examination of a new (3+1)-dimensional fifth-order nonlinear Wazwaz equation with third-order dispersion terms which is applicable in ocean physics and other nonlinear sciences. Ocean physics is the study of various conditions as well as processes that are physical within the ocean, especially the physical features and motions of ocean waters. Thus, the diverse outcomes of the investigations carried out on the model under study possesses significant applications in physical oceanography (Adeyemo, 2022). We first apply Lie group analysis to obtain various infinitesimal generators admitted by the equation. This is followed by invoking the generators to reduce the understudy equation for the purpose of achieving copious group-invariant solutions. We solve the resultant ordinary differential equations in order to obtain diverse classical solutions of the underlying equation. The outcome yields several solutions with arbitrary functions, quadratures, rational and hyperbolic functions alongside various topological kink solitons. Moreover, numerical simulations of the obtained results are performed so as to give some physical interpretations to the solutions and these yield various solitary waves as well as interactions between soliton waves of interest. Furthermore, we outline the applications of our results in ocean physics. Conclusively, conservation laws are constructed for the equation understudy with the use of Ibragimov’s theorem.