Approximate Symmetry Reduction Research Articles

Approximate symmetry reduction for initial-value problem of perturbed diffusion equations

In this paper, the approximate symmetry reduction for the initial-value problem of perturbed diffusion equations with source term is studied by the approximate generalized conditional symmetry. The classification of governing equations is given, and the Cauchy problem of partial differential equations is reduced to initial-value problem of ordinary differential equations. Finally, the approximate solution is obtained by solving the reduced system of equations.

Approximate symmetry reduction to the perturbed coupled KdV equations derived from two-layer fluids

Purpose: The purpose of this paper is to research the solution of the perturbed coupled Korteweg-de Vries (KdV) equations. Methods: The results are obtained through researching a system of corresponding partial differential equations which approximates the perturbed coupled KdV equations and the symmetry method. Results: The symmetry structure and the corresponding general approximate symmetry reduction are derived. Then, using the symmetry reduction, the infinite series solutions are obtained. Conclusions: The approximate symmetry reduction approach can be used to search for similar results of other perturbed nonlinear differential equations with small parameters.

Approximate Symmetry Reduction for Initial-value Problems of the Extended KdV-Burgers Equations with Perturbation

Approximate generalized conditional symmetry is developed to study the approximate symmetry reduction for initial-value problems of the extended KdV-Burgers equations with perturbation. These equations can be reduced to initial-value problems for some systems of first-order perturbed ordinary differential equations in terms of a new approach. Complete classification theorems are obtained and an example is taken to show the main reduction procedure.

The approximate homotopy symmetry reduction for far-field model equation

The far-field model equation is investigated by the approximate homotopy symmetry method. Homotopy series solutions are constructed through summarizing the relevant general formulas for similarity reduction solutions and similarity reduction equations of differernt orders. Similarity reduction equations of different orders are linear variable coefficients ordinary differential equations, and can be solved one by one from zero-order similarity reduction equations. The auxiliary parameter in the homotopy model affects convergence of homotopy series solutions.

Approximate Similarity Reduction for the Nonlinear K(n, 1) Equation with Weak Damping via Symmetry Perturbation and Direct Method

The nonlinear K(n, 1) equation with weak damping is investigated via the approximate symmetry perturbation method andapproximate direct method. The approximate symmetry and similarity reduction equations of different orders are derivedand the corresponding series reduction solutions are obtained. As a result, the formal coincidence for both methods isdisplayed.

Approximate Homotopy Symmetry Reduction Method: Infinite Series Reductions to Kawahara Equation

The Kawahara equation is studied through the approximate homotopy symmetry method. Under this method we get the similarity reduction solutions of the Kawahara equation, leading to the corresponding homotopy series solutions. Furthermore, the similarity solutions of the corresponding reduced linear ordinary differential equations are also considered.

Approximate symmetry reduction for perturbed nonlinear Schrödinger equation

We investigate the one-dimensional nonlinear Schrödinger equation with a perturbation of polynomial type. The approximate symmetries and approximate symmetry reduction equations are obtained with the approximate symmetry perturbation theory.

A lazy approach to symmetry reduction

Abstract Symmetry reduction is a technique to counter state explosion for systems with regular structure. It relies on idealistic assumptions about indistinguishable components, which in practice may only be similar. In this article, we present a flexible, lazy approach to symmetry-reducing a structure without any prior knowledge about its global symmetry. Instead of a-priori checking for compliance with symmetry conditions, each encountered state is annotated on the fly with information about how symmetry is violated along the path leading to it. The method naturally favors “very symmetric” systems: more similarity among the components leads to greater compression. A notion of subsumption is used to prune the annotated search space during exploration. Previous solutions to the approximate symmetry reduction problem are restricted to specific types of asymmetry, such as up to bisimilarity, or incur a large overhead, either during preprocessing of the structure or during the verification run. In contrast, the strength of our method is its balance between ease of implementation and algorithmic flexibility. We include analytic and experimental results that witness its efficiency.

Approximate Symmetry Reduction Approach: Infinite Series Reductions to the KdV-Burgers Equation

For weak dispersion and weak dissipation cases, the (1+1)-dimensional KdV-Burgers equation is investigated in terms of approximate symmetry reduction approach. The formal coherence of similarity reduction solutions and similarity reduction equations of different orders enables series reduction solutions. For the weak dissipation case, zero-order similarity solutions satisfy the Painlevé II, Painlevé I, and Jacobi elliptic function equations. For the weak dispersion case, zero-order similarity solutions are in the form of Kummer, Airy, and hyperbolic tangent functions. Higher-order similarity solutions can be obtained by solving linear variable coefficients ordinary differential equations.

Approximate Symmetry Reduction and Infinite Series Solutions to the Nonlinear Wave Equation with Damping

The approximate symmetry perturbation method is applied to the nonlinear damped wave equation. The approximate symmetry reduction equations of different orders are derived and the corresponding series reduction solutions are obtained.

Approximate Symmetries and Infinite Series Symmetry Reduction Solutions to Perturbed Kuramoto–Sivashinsky Equation

Starting from Lie symmetry theory and combining with the approximate symmetry method, and using the package LieSYMGRP proposed by us, we restudy the perturbed Kuramoto–Sivashinsky (KS) equation. The approximate symmetry reduction and the infinite series symmetry reduction solutions of the perturbed KS equation are constructed. Specially, if selecting the tanh-type travelling wave solution as initial approximate, we not only obtain the general formula of the physical approximate similarity solutions, but also obtain several new explicit solutions of the given equation, which are first reported here.

Infinite series symmetry reduction solutions to the modified KdV–Burgers equation

From the point of view of approximate symmetry, the modified Korteweg-de Vries–Burgers (mKdV–Burgers) equation with weak dissipation is investigated. The symmetry of a system of the corresponding partial differential equations which approximate the perturbed mKdV–Burgers equation is constructed and the corresponding general approximate symmetry reduction is derived; thereby infinite series solutions and general formulae can be obtained. The obtained result shows that the zero-order similarity solution to the mKdV–Burgers equation satisfies the Painlevé II equation. Also, at the level of travelling wave reduction, the general solution formulae are given for any travelling wave solution of an unperturbed mKdV equation. As an illustrative example, when the zero-order tanh profile solution is chosen as an initial approximate solution, physically approximate similarity solutions are obtained recursively under the appropriate choice of parameters occurring during computation.

Approximate Symmetry Reduction to the Perturbed One-Dimensional Nonlinear Schrödinger Equation

The one-dimensional nonlinear Schrödinger equation with a perturbation of polynomial type is considered. Using the approximate symmetry perturbation theory, the approximate symmetries and approximate symmetry reduction equations are obtained.