In this manuscript, a new form of the generalized q-deformed Sinh-Gordon equation is investigated which could model physical systems with broken symmetries and to incorporate phenomena involving amplification or dissipation. The proposed model is explored based on the Lie symmetry approach. Using similarity reduction, the partial differential equation is transformed into an ordinary differential equation. By employing the generalized auxiliary equation approach, precise results for the derived equation are obtained. The solutions are graphically depicted as 3D, 2D, and contour plots. Furthermore, the qualitative analysis of the considered model is investigated by employing the concepts of bifurcation and chaos. The phase profiles are displayed for different sets of the parameters. Additionally, by applying an external periodic strength, quasi-periodic and chaotic behaviors are documented. Various tools for detecting chaos are discussed, including 3D and 2D phase patterns, time series, and Poincaré maps. Additionally, a sensitivity analysis is conducted for various initial conditions. The obtained findings are unique and indicate the viability and efficacy of the suggested strategies for evaluating soliton solutions and phase illustrations for various nonlinear models.

/pdf/the-analysis-of-bifurcation-quasi-periodic-and-solitons-3muq8wym.pdf

The Analysis of Bifurcation, Quasi-Periodic and Solitons Patterns to the New Form of the Generalized q-Deformed Sinh-Gordon Equation

Symmetry- and conservation law-preserving finite difference discretizations are obtained for linear and nonlinear one-dimensional wave equations on five- and nine-point stencils using the theory of Lie point symmetries of difference equations and the discrete direct multiplier method of conservation law construction. In particular, for the linear wave equation, an explicit five-point scheme is presented that preserves the discrete analogs of its basic geometric point symmetries and six of the corresponding conservation laws. For a class of nonlinear wave equations arising in hyperelasticity, a nine-point implicit scheme is constructed, preserving four-point symmetries and three local conservation laws. Other discretizations of the nonlinear wave equations preserving different subsets of conservation laws are discussed.

Invariant conservation law-preserving discretizations of linear and nonlinear wave equations

The one-dimensional shallow water equations in Eulerian coordinates are considered. Relations between symmetries and conservation laws for the potential form of the equations and symmetries and conservation laws in Eulerian coordinates are shown. An invariant difference scheme for equations in Eulerian coordinates with arbitrary bottom topography is constructed. It possesses all the finite-difference analogs of the conservation laws. Some bottom topographies require moving meshes in Eulerian coordinates, which are stationary meshes in mass Lagrangian coordinates. The developed invariant conservative difference schemes are verified numerically using examples of flow with various bottom topographies.

Discrete shallow water equations preserving symmetries and conservation laws

The main purpose of this article is to show how structure reected in partial dierential equations can be preserved in a discrete world and reected in dierence schemes. Three dierent structure preserving discretizations of the Liouville equation are presented and then used to solve specic boundary value problems. The results are compared with exact solutions satisfying the same boundary conditions. All three discretizations are on four point lattices. One preserves linearizability of the equation, another the innite dimensional symmetry group as higher symmetries, the third preserves the maximal nite dimensional subgroup of the symmetry group as point symmetries. A 9-point invariant scheme that gives a better approximation of the equation, but worse numerical results for solutions is presented and discussed.

/pdf/structure-preserving-discretizations-of-the-liouville-1ksmuajsg3.pdf

Structure preserving discretizations of the Liouville equation and their numerical tests

We provide the Hamiltonian version of the approximate Noether theorem developed for the perturbed ordinary differential equations (ODEs) (Govinder et al. in Phys Lett 240(3):127–131, 1998) for the approximate Hamiltonian systems. We follow the procedure adopted by Dorodnitsyn and Kozlov (J Eng Math 66(1–3):253–270, 2010) for the Hamiltonian systems of unperturbed ODEs. The approximate Legendre transformation connects the approximate Hamiltonian and approximate Lagrangian. The approximate Noether symmetries determining equation for the approximate Hamiltonian systems is defined explicitly. We provide a formula to establish an approximate first integral associated with an approximate Noether symmetry of the approximate Hamiltonian systems. We analyzed several physical models to elaborate the approach developed here.

The approximate Noether symmetries and approximate first integrals for the approximate Hamiltonian systems

A recent paper considered symmetries and conservation laws of the plane one-dimensional flows for magnetohydrodynamics in the mass Lagrangian coordinates. This paper analyses the one-dimensional magnetohydrodynamics flows with cylindrical symmetry in the mass Lagrangian coordinates. The medium is assumed inviscid and thermally non-conducting. It is modeled by a polytropic gas. Symmetries and conservation laws are found. The cases of finite and infinite electric conductivity need to be analyzed separately. For finite electric conductivity $\sigma (\rho,p)$ we perform Lie group classification, which identifies $\sigma (\rho,p)$ cases with additional symmetries. The conservation laws are found by direct computation. For cases with infinite electric conductivity variational formulations of the equations are considered. Lie group classifications are obtained with the entropy treated as an arbitrary element. A variational formulation allows to use the Noether theorem for computation of conservation laws. The conservation laws obtained for the variational equations are also presented in the original (physical) variables. 

pdf/one-dimensional-mhd-flows-with-cylindrical-symmetry-lie-3g9ggrhp.pdf

One-dimensional MHD flows with cylindrical symmetry: Lie symmetries and conservation laws

The paper presents a new method for finding first integrals of difference equations which do not possess Lagrangians, nor Hamiltonians. In this paper we consider ordinary differential and difference equations. As an example we solve a third order nonlinear ordinary differential equation and its invariant discretization using three first integrals obtained by means of this method.

First integrals of difference equations which do not possess a variational formulation

Invariant finite-difference schemes are considered for one-dimensional magnetohydrodynamics (MHD) equations in mass Lagrangian coordinates for the cases of finite and infinite conductivity. The construction of these schemes makes use of results of the group classification of MHD equations previously obtained by the authors. On the basis of the classical Samarskiy–Popov scheme, new schemes are constructed for the case of finite conductivity. These schemes admit all symmetries of the original differential model and have difference analogues of all of its local differential conservation laws. New, previously unknown, conservation laws are found using symmetries and direct calculations. In the case of infinite conductivity, conservative invariant schemes are constructed as well. For isentropic flows of a polytropic gas the proposed schemes possess the conservation law of energy and preserve entropy on two time layers. This is achieved by means of specially selected approximations for the equation of state of a polytropic gas. In addition, invariant difference schemes with additional conservation laws are proposed. A new scheme for the case of finite conductivity is tested numerically for various boundary conditions, which shows accurate preservation of difference conservation laws.

/pdf/invariant-finite-difference-schemes-for-plane-one-156v0pxz.pdf

Invariant Finite-Difference Schemes for Plane One-Dimensional MHD Flows That Preserve Conservation Laws

Symmetries of the one-dimensional shallow water magnetohydrodynamics equations (SMHD) in Gilman’s approximation are studied. The SMHD equations are considered in case of a plane and uneven bottom topography in Lagrangian and Eulerian coordinates. Symmetry classification separates out all bottom topographies which yields substantially different admitted symmetries. The SMHD equations in Lagrangian coordinates were reduced to a single second order PDE. The Lagrangian formalism and Noether’s theorem are used to construct conservation laws of the SMHD equations. Some new conservation laws for various bottom topographies are obtained. The results are also represented in Eulerian coordinates. Invariant and partially invariant solutions are constructed.

/pdf/symmetries-and-conservation-laws-of-the-one-dimensional-23yu0s2y.pdf

Symmetries and conservation laws of the one-dimensional shallow water magnetohydrodynamics equations in Lagrangian coordinates

The paper analyzes one of the models of equations of magnetohydrodynamics (MHD) derived earlier. The model was obtained as a result of group classification of the MHD equations in mass Lagrangian coordinates, where all dependent variables in Eulerian coordinates depend on time and two spatial coordinates. The use of Lagrangian coordinates made it possible to solve four equations, which led to the form of reduced equations containing four arbitrary functions: entropy and a three-dimensional vector associated with the magnetic field. The objective of this work is to develop conservation laws and exact solutions for the model. Conservation laws are obtained using Noether's theorem, while exact solutions are obtained either explicitly or by solving a system of ordinary or partial differential equations with two independent variables. Numerical methods are employed for the latter solutions. 

One class of MHD equations: Conservation laws and exact solutions

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