Lie symmetries of geodesic equations and projective collineations

We prove a theorem which relates the Lie symmetries of the geodesic equations in a Riemannian space with the collineations of the metric. We apply the results to Einstein spaces and spaces of constant curvature. Finally with examples we show the use of the results.     

Lie symmetries,integrable properties and exact solutions to the variable-coefficient nonlinear evolution equations

Nonlinear Dynamics - Mechanism for the generation of attractor in nonlinear systems with hidden attractor is still not understood completely. Since such systems do not possess all the requisite...     

Exact solutions to magnetogasdynamics using Lie point symmetries

In the present work, we find some exact solutions to the first order quasilinear hyperbolic system of partial differential equations (PDEs), governing the one dimensional unsteady flow of inviscid and perfectly conducting compressible fluid, subjected to a transverse magnetic field. For this, Lie group analysis is used to identify a finite number of generators that leave the given system of PDEs invariant. Out of these generators, two commuting generators are constructed involving some arbitrary constants. With the help of canonical variables associated with these two generators, the assigned system of PDEs is reduced to an autonomous system whose simple solutions provide nontrivial solutions of the original system. Using this exact solution, we discuss the evolutionary behavior of weak discontinuities.     

Two new integrable fourth-order nonlinear equations: multiple soliton solutions and multiple complex soliton solutions

Nonlinear Dynamics - In this paper, we develop two new fourth-order integrable equations represented by nonlinear PDEs of second-order derivative in time t. The new equations model both right- and...     

On the stability of plane wave and soliton solutions to the Zakharov equations

Plane wave and soliton solutions of the two types of Zakharov equation (two dimensional and simplified one directional) are considered. Stability properties in one dimensional space are seen to be similar. This is interesting, as the first type of equation is not solvable whereas the second is. The soliton solutions of both are one dimensionally stable but those of the full Zakharov equations are unstable with respect to perpendicular perturbations. Regions of stability of nonlinear wave and shock wave solutions in parameter space as well as growth rates of instabilities are given.     

Asymptotic soliton train solutions of Kaup–Boussinesq equations

Asymptotic soliton trains arising from a ‘large and smooth’ enough initial pulse are investigated by the use of the quasiclassical quantization method for the case of Kaup–Boussinesq shallow water equations. The parameter varying along the soliton train is determined by the Bohr–Sommerfeld quantization rule which generalizes the usual rule to the case of ‘two potentials’ h₀(x) and u₀(x) representing initial distributions of height and velocity, respectively. The influence of the initial velocity u₀(x) on the asymptotic stage of the evolution is determined. Excellent agreement of numerical solutions of the Kaup–Boussinesq equations with predictions of the asymptotic theory is found.     

Lie symmetries and conserved quantities of rotational relativistic systems

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Nonlocal symmetries of evolution equations

We suggest the method for group classification of evolution equations admitting nonlocal symmetries which are associated with a given evolution equation possessing nontrivial Lie symmetry. We apply this method to second-order evolution equations in one spatial variable invariant under Lie algebras of the dimension up to three. As a result, we construct the broad families of new nonlinear evolution equations possessing nonlocal symmetries which in principle cannot be obtained within the classical Lie approach.     

Dynamics of some novel breather solutions and rogue waves for the PT-symmetric nonlocal soliton equations

Nonlinear Dynamics - Tuned mass dampers are useful devices for limiting vibrations in various machines. Their main advantage is that they can be used as add-on elements and do not require...     

On symmetries, conservation laws and invariant solutions of the foam-drainage equation

This study deals with symmetry group properties and conservation laws of the foam-drainage equation. Firstly, we study the classical Lie symmetries, optimal systems, similarity reductions and similarity solutions of the foam-drainage equation which are obtained through the Lie group method of infinitesimal transformations. Secondly, using the new general theorem on non-local conservation laws and partial Lagrangian approach, local and non-local conservation laws are also studied and, finally, non-classical symmetries are derived.     

Smooth soliton and kink solutions for a new integrable soliton equation

Nonlinear Dynamics - We establish a relationship between a new integrable soliton equation and Gardner’s equation by a transformation. Then, we use this transformation and solutions of...     

Exact solutions to the equations of perfect gases through Lie group analysis and substitution principles

In this paper we consider the equations that govern the motion of perfect gases. We explicitly characterize some classes of steady solutions in two and three space dimensions, by introducing invertible point transformations suggested by Lie group analysis; moreover, by using various transformations known as substitution principles, new steady and unsteady solutions are constructed.     

On global solutions of functional differential equations

We construct a system of linear differential equations all solutions of which are global solutions of a system of functional differential equations. We substantiate the existence of this system and examine some properties of its solutions.     

Approximate symmetries and solutions of the Kompaneets equation

Different approximations of the Kompaneets equation are studied using approximate symmetries, which allows consideration of the contributions of all terms of this equation previously neglected in the analysis of the limiting cases.     

Exact solutions to the unsteady equations of perfect gases through Lie group analysis and substitution principles

In this paper, we consider the unsteady equations that govern two- and three-dimensional flows of a perfect gas. We explicitly characterize various classes of exact solutions by introducing some invertible transformations suggested by the invariance with respect to Lie groups of point symmetries and using suitable transformations known in literature as substitution principles.     

On the N-wave equations and soliton interactions in two and three dimensions

Several important examples of the N-wave equations are studied. These integrable equations can be linearized by formulation of the inverse scattering as a local Riemann-Hilbert problem (RHP). Several nontrivial reductions are presented. Such reductions can be applied to the generic N-wave equations but mainly the 3- and 4-wave interactions are presented as examples. Their one and two-soliton solutions are derived and their soliton interactions are analyzed. It is shown that additional reductions may lead to new types of soliton solutions. In particular the 4-wave equations with ?₂ × ?₂ reduction group allow breather-like solitons. Finally it is demonstrated that RHP with sewing function depending on three variables t, x and y provides some special solutions of the N-wave equations in three dimensions.