Applications of Solvable Lie Algebras to a Class of Third Order Equations

A family of third-order partial differential equations (PDEs) is analyzed. This family broadens out well-known PDEs such as the Korteweg-de Vries equation, the Gardner equation, and the Burgers equation, which model many real-world phenomena. Furthermore, several macroscopic models for semiconductors considering quantum effects—for example, models for the transmission of electrical lines and quantum hydrodynamic models—are governed by third-order PDEs of this family. For this family, all point symmetries have been derived. These symmetries are used to determine group-invariant solutions from three-dimensional solvable subgroups of the complete symmetry group, which allow us to reduce the given PDE to a first-order nonlinear ordinary differential equation (ODE). Finally, exact solutions are obtained by solving the first-order nonlinear ODEs or by taking into account the Type-II hidden symmetries that appear in the reduced second-order ODEs.

Lie symmetry analysis, exact solutions, conservation laws of variable-coefficients Calogero–Bogoyavlenskii–Schiff equation

In this paper, a [Formula: see text]-dimensional variable-coefficients Calogero–Bogoyavlenskii–Schiff (vcCBS) equation is studied. The infinitesimal generators and symmetry groups are obtained by using the Lie symmetry analysis on vcCBS. The optimal system of one-dimensional subalgebras of vcCBS is computed for determining the group-invariant solutions. On this basis, the vcCBS is reduced to two-dimensional partial differential equations (PDEs) by similarity reductions. Furthermore, the reduced PDEs are solved to obtain the two-soliton interaction solution, the soliton-kink interaction solution and some other exact solutions by the [Formula: see text]-expansion method. Moreover, it is shown that vcCBS is nonlinearly self-adjoint and then its conservation laws are calculated.

Symmetry Solutions and Conservation Laws for the 3D Generalized Potential Yu-Toda-Sasa-Fukuyama Equation of Mathematical Physics

In this paper we study the fourth-order three-dimensional generalized potential Yu-Toda-Sasa-Fukuyama (gpYTSF) equation by first computing its Lie point symmetries and then performing symmetry reductions. The resulting ordinary differential equations are then solved using direct integration, and exact solutions of gpYTSF equation are obtained. The obtained group invariant solutions include the solution in terms of incomplete elliptic integral. Furthermore, conservation laws for the gpYTSF equation are derived using both the multiplier and Noether’s methods. The multiplier method provides eight conservation laws, while the Noether’s theorem supplies seven conservation laws. These conservation laws include the conservation of energy and mass.

Two Isospectral-Nonisospectral Super-Integrable Hierarchies and Related Invariant Solutions

In this article, we adopt two kinds of loop algebras corresponding to the Lie algebra B(0,1) to introduce two line spectral problems with different numbers of even and odd superfunctions. Through generalizing the time evolution λt to a polynomial of λ, two isospectral-nonisospectral super integrable hierarchies are derived in terms of Tu scheme and zero-curvature equation. Among them, the first super integrable hierarchy is further reduced to generalized Fokker–Plank equation and special bond pricing equation, as well as an explicit super integrable system under the choice of specific parameters. More specifically, a super integrable coupled equation is derived and the corresponding integrable properties are discussed, including the Lie point symmetries and one-parameter Lie symmetry groups as well as group-invariant solutions associated with characteristic equation.

A NEW CONSTRUCTION OF OS OF SUBALGEBRAS AND INVARIANT SOLUTION OF THE BLACK-SCHOLES EQUATION

In this manuscript, the Lie group technique is applied to construct a new OS and invariant solutions of a one-dimensional LA, which describes the symmetries properties of a nonlinear Black-Scholes model. The structure of LA depends on one parameter. We have shown a novel way to construct the so-called OS of subalgebras of the Black-Scholes equation by utilizing the given symmetries. We transform the symmetries of the Black-Scholes equation into a simple ordinary differential equation called the Lie equation, which provides us a way through which to construct a new optimal scheme of subalgebras of the Black-Scholes through applying the concept of LE. The OS which consists of minimal representatives is utilized to develop the invariant solution for the Black-Scholes equation. The fundamental use of the Lie group analysis to the differential equation is the categorization of group invariant solutions of differential equations via OS. Finally, we have utilized the OS to construct the invariant solution of the Black-Scholes equation.

Residual Symmetry, Bäcklund Transformation, and Soliton Solutions of the Higher-Order Broer-Kaup System

Under investigation in this paper is the higher-order Broer-Kaup(HBK) system, which describes the bidirectional propagation of long waves in shallow water. Via the standard truncated Painlevé expansion method, the residual symmetry of this system is derived. By introducing an appropriate auxiliary-dependent variable, the residual symmetry is successfully localized to Lie point symmetries. Via solving the initial value problems, the finite symmetry transformations are presented. However, the solution which obtained from the residual symmetry is a special group invariant solutions. In order to find more general solution of HBK system, we further generalize the residual symmetry method to the consistent tanh expansion (CTE) method and prove that the HBK system is CTE solvable, then the resonant soliton solutions and interaction solutions among different nonlinear excitations are obtained by the CET method.

Symmetries and Invariant Solutions for the Coagulation of Aerosols

The coagulation of aerosol particles plays an important role in the structural morphological changes of suspended particles at any time and in any space. In this study, based on the Smoluchowski equation of population balance, a kinetic model of aerosol coalescence considering Brownian motion collision is established. By applying the developed Lie group method, we derive the allowed infinitesimal symmetries and group-invariant solutions of the integro-differential equation, as well as the exact solution under some special conditions. We also provide detailed steps and a discussion of the properties. The content and results provide an effective analytic solution for the progressive evolution of aerosol particle size considering boundary and initial conditions. This solution reveals the self-conservative phenomena in the process of aerosol coalescence and also provides validation for the numerical algorithms of general dynamics equations.