Exact solutions and aysmptotic state analysis of a modified Toda lattice system with a perturbation parameter

Under consideration is a modified Toda lattice system with a perturbation parameter, which may describe the particle motion in a lattice. With the aid of symbolic computation Maple, the discrete generalized [Formula: see text]-fold Darboux transformation (DT) of this system is constructed for the first time. Different types of exact solutions are derived by applying the resulting DT through choosing different [Formula: see text]. Specifically, standard soliton solutions, rational solutions and their mixed solutions are given via the [Formula: see text]-fold DT, [Formula: see text]-fold DT and [Formula: see text]-fold DT, respectively. Limit states of various exact solutions are analyzed via the asymptotic analysis technique. Compared with the known results, we find that the asymptotic states of mixed solutions of standard soliton and rational solutions are consistent with the asymptotic analysis results of solitons and rational solutions alone. Soliton interaction and propagation phenomena are shown graphically. Numerical simulations are used to explore relevant soliton dynamical behaviors. These results and properties might be helpful for understanding lattice dynamics.

Hybrid high-order method for singularly perturbed fourth-order problems on curved domains

We propose a novel hybrid high-order method (HHO) to approximate singularly perturbed fourth-order PDEs on domains with a possibly curved boundary. The two key ideas in devising the method are the use of a Nitsche-type boundary penalty technique to weakly enforce the boundary conditions and a scaling of the weighting parameter in the stabilisation operator that compares the singular perturbation parameter to the square of the local mesh size. With these ideas in hand, we derive stability and optimal error estimates over the whole range of values for the singular perturbation parameter, including the zero value for which a second-order elliptic problem is recovered. Numerical experiments illustrate the theoretical analysis.

First-order perturbations of Gödel-type metrics in non-dynamical Chern-Simons modified gravity

G\"{o}del-type metrics that are homogeneous in both space and time remain, like the Schwarzschild metric, consistent within Chern-Simons modified gravity; this is true in both the non-dynamical and dynamical frameworks, each of which involves an additional pseudoscalar field coupled to the Pontryagin density. In this paper, we consider stationary first-order perturbations to these metrics in the non-dynamical framework. Under certain assumptions we find analytical solutions to the perturbed field equations. The solutions of the first-order field equations break the translational and cylindrical symmetries of the unperturbed metrics. The effective potential controlling planar geodesic orbits is also affected by the perturbation parameter, which changes the equilibrium radii for the orbits of both massive particles and massless photons.

Low-rank tensor approximation of singularly perturbed boundary value problems in one dimension

We derive rank bounds on the quantized tensor train (QTT) compressed approximation of singularly perturbed reaction diffusion boundary value problems in one dimension. Specifically, we show that, independently of the scale of the singular perturbation parameter, a numerical solution with accuracy $$0<\varepsilon <1$$ 0 < ε < 1 can be represented in the QTT format with a number of parameters that depends only polylogarithmically on $$\varepsilon $$ ε . In other words, QTT-compressed solutions converge exponentially fast to the exact solution, with respect to a root of the number of parameters. We also verify the rank bound estimates numerically and overcome known stability issues of the QTT-based solution of partial differential equations (PDEs) by adapting a preconditioning strategy to obtain stable schemes at all scales. We find, therefore, that the QTT-based strategy is a rapidly converging algorithm for the solution of singularly perturbed PDEs, which does not require prior knowledge on the scale of the singular perturbation and on the shape of the boundary layers.

Existence results for a nonlinear nonautonomous transmission problem via domain perturbation

In this paper we study the existence and the analytic dependence upon domain perturbation of the solutions of a nonlinear nonautonomous transmission problem for the Laplace equation. The problem is defined in a pair of sets consisting of a perforated domain and an inclusion whose shape is determined by a suitable diffeomorphism $\phi$ . First we analyse the case in which the inclusion is a fixed domain. Then we will perturb the inclusion and study the arising boundary value problem and the dependence of a specific family of solutions upon the perturbation parameter $\phi$ .

Numerical Method of the Line for Solving One Dimensional Initial- Boundary Singularly Perturbed Burger Equation

In this Research Method of Line is used to find the approximation solution of one dimensional singularly perturbed Burger equation given with initial and boundary conditions. First, the given solution domain is discretized and the derivative involving the spatial variable x is replaced into the functional values at each grid points by using the central finite difference method. Then, the resulting first-order linear ordinary differential equation is solved by the fifth-order Runge-Kutta method. To validate the applicability of the proposed method, one model example is considered and solved for different values of the perturbation parameter ‘ε’ and mesh sizes in the direction of the temporal variable, t. Numerical results are presented in tables in terms of Maximum point-wise error, EN,Δt and rate of convergence, Pε N,Δt. The stability of this new class of Numerical method is also investigated by using Von Neumann stability analysis techniques. The numerical results presented in tables and graphs confirm that the approximate solution is in good agreement with the exact solution.

Numerical Method of the Line for Solving One Dimensional Initial- Boundary Singularly Perturbed Burger Equation

In this Research Method of Line is used to find the approximation solution of one dimensional singularly perturbed Burger equation given with initial and boundary conditions. First, the given solution domain is discretized and the derivative involving the spatial variable x is replaced into the functional values at each grid points by using the central finite difference method. Then, the resulting first-order linear ordinary differential equation is solved by the fifth-order Runge-Kutta method. To validate the applicability of the proposed method, one model example is considered and solved for different values of the perturbation parameter ‘  ’ and mesh sizes in the direction of the temporal variable, t. Numerical results are presented in tables in terms of Maximum point-wise error, N t , E  and rate of convergence, N t , P  . The stability of this new class of Numerical method is also investigated by using Von Neumann stability analysis techniques. The numerical results presented in tables and graphs confirm that the approximate solution is in good agreement with the exact solution.

Existence and Stability of Kayaking Orbits for Nematic Liquid Crystals in Simple Shear Flow

We use geometric methods of equivariant dynamical systems to address a long-standing open problem in the theory of nematic liquid crystals, namely a proof of the existence and asymptotic stability of kayaking periodic orbits in response to steady shear flow. These are orbits for which the principal axis of orientation of the molecular field (the director) rotates out of the plane of shear and around the vorticity axis. With a small parameter attached to the symmetric part of the velocity gradient, the problem can be viewed as a symmetry-breaking bifurcation from an orbit of the rotation group $$\mathrm{SO}(3)$$ SO ( 3 ) that contains both logrolling (equilibrium) and tumbling (periodic rotation of the director within the plane of shear) regimes as well as a continuum of neutrally stable kayaking orbits. The results turn out to require expansion to second order in the perturbation parameter.

Calmness of Linear Constraint Systems under Structured Perturbations with an Application to the Path-Following Scheme

We are concerned with finite linear constraint systems in a parametric framework where the right-hand side is an affine function of the perturbation parameter. Such structured perturbations provide a unified framework for different parametric models in the literature, as block, directional and/or partial perturbations of both inequalities and equalities. We extend some recent results about calmness of the feasible set mapping and provide an application to the convergence of a certain path-following algorithmic scheme. We underline the fact that our formula for the calmness modulus depends only on the nominal data, which makes it computable in practice.

Fitted Numerical Scheme for Second-Order Singularly Perturbed Differential-Difference Equations with Mixed Shifts

This paper presents the study of singularly perturbed differential-difference equations of delay and advance parameters. The proposed numerical scheme is a fitted fourth-order finite difference approximation for the singularly perturbed differential equations at the nodal points and obtained a tridiagonal scheme. This is significant because the proposed method is applicable for the perturbation parameter which is less than the mesh size , where most numerical methods fail to give good results. Moreover, the work can also help to introduce the technique of establishing and making analysis for the stability and convergence of the proposed numerical method, which is the crucial part of the numerical analysis. Maximum absolute errors range from 10 − 03 up to 10 − 10 , and computational rate of convergence for different values of perturbation parameter, delay and advance parameters, and mesh sizes are tabulated for the considered numerical examples. Concisely, the present method is stable and convergent and gives more accurate results than some existing numerical methods reported in the literature.