A Study of (2+1)-Dimensional Konopelchenko-Dubrovsky (KD) System: Closed-Form Solutions, Solitary Waves, Bifurcation Analysis and Quasi-Periodic Solution

Nonlinear evolution equations (NLEEs) are extensively used to establish the elementary propositions of natural circumstances. In this work, we study the Konopelchenko-Dubrovsky (KD) equation which depicts non-linear waves in mathematical physics with weak dispersion. The considered model is investigated using the combination of generalized exponential rational function (GERF) method and dynamical system method. The GERF method is utilized to generate closedform invariant solutions to the (2+1)-dimensional KD model in terms of trigonometric, hyperbolic, and exponential forms with the assistance of symbolic computations. Moreover, three-dimensional graphics are displayed to depict the behavior of obtained solitary wave solutions. The model is observed to have single and multiple soliton profiles, kink-wave profiles, and periodic oscillating nonlinear waves. These generated solutions have never been published in the literature. All the newly generated soliton solutions are checked by putting them back into the associated system with the soft computation via Wolfram Mathematica. Moreover, the system is converted into a planer dynamical system using a certain transformation and the analysis of bifurcation is examined. Furthermore, the quasi-periodic solution is investigated numerically for the perturbed system by inserting definite periodic forces into the considered model. With regard to the parameter of the perturbed model, two-dimensional and three-dimensional phase portraits are plotted.

No-go theorem for inflation in an extended Ricci-inverse gravity model

In this paper, we propose an extension of the Ricci-inverse gravity, which has been proposed recently as a very novel type of fourth-order gravity, by introducing a second order term of the so-called anticurvature scalar as a correction. The main purpose of this paper is that we would like to see whether the extended Ricci-inverse gravity model admits the homogeneous and isotropic Friedmann–Lemaitre–Robertson–Walker metric as its stable inflationary solution. However, a no-go theorem for inflation in this extended Ricci-inverse gravity is shown to appear through a stability analysis based on the dynamical system method. As a result, this no-go theorem implies that it is impossible to have such stable inflation in this extended Ricci-inverse gravity model.

Separation Method of Semi-Fixed Variables Together with Dynamical System Method for Solving Nonlinear Time-Fractional PDEs with Higher-Order Terms

It is well known that methods for solving fractional-order PDEs are grossly inadequate compared with integer-order PDEs. In this paper, a new approach which combined with the separation method of semi-fixed variables and dynamical system method is introduced. As example, a time-fractional reaction-diffusion equation with higher-order terms is studied under the different kinds of fractional-order differential operators. In different parametric regions, phase portraits of systems which derived from the reaction-diffusion equation are presented. Existence and dynamic properties of solutions of this nonlinear time-fractional models are investigated. In some special parametric conditions, some exact solutions of this time-fractional models are obtained. The dynamical properties of some exact solutions are discussed and the graphs of them are illustrated.PACS: 02.30.Jr; 02.30.Oz; 02.70.-c; 02.70.Mv; 02.90.+p; 04.20.Jb; 05.10.-a

Anisotropic power-law inflation for a model of two scalar and two vector fields

Inspired by an interesting counterexample to the cosmic no-hair conjecture found in a supergravity-motivated model recently, we propose a multi-field extension, in which two scalar fields are allowed to non-minimally couple to two vector fields, respectively. This model is shown to admit an exact Bianchi type I power-law solution. Furthermore, stability analysis based on the dynamical system method is performed to show that this anisotropic solution is indeed stable and attractive if both scalar fields are canonical. Nevertheless, if one of the two scalar fields is phantom then the corresponding anisotropic power-law inflation turns unstable as expected.

No-go theorem for inflation in Ricci-inverse gravity

In this paper, we study the so-called Ricci-inverse gravity, which is a very novel type of fourth-order gravity proposed recently. In particular, we are able to figure out both isotropically and anisotropically inflating universes to this model. More interestingly, these solutions are shown to be free from a singularity problem. However, stability analysis based on the dynamical system method shows that both isotropic and anisotropic inflation of this model turn out to be unstable against field perturbations. This result implies a no-go theorem for both isotropic and anisotropic inflation in the Ricci-inverse gravity.

Stable small spatial hairs in a power-law k-inflation model

In this paper, we extend our investigation of the validity of the cosmic no-hair conjecture within non-canonical anisotropic inflation. As a result, we are able to figure out an exact Bianchi type I solution to a power-law k-inflation model in the presence of unusual coupling between scalar and electromagnetic fields as $$-f^2(\phi )F_{\mu \nu }F^{\mu \nu }/4$$ - f 2 ( ϕ ) F μ ν F μ ν / 4 . Furthermore, stability analysis based on the dynamical system method indicates that the obtained solution does admit stable and attractive hairs during an inflationary phase and therefore violates the cosmic no-hair conjecture. Finally, we show that the corresponding tensor-to-scalar ratio of this model turns out to be highly consistent with the observational data of the Planck 2018.

Travelling Wave Solutions of Wu–Zhang System via Dynamic Analysis

In this paper, based on the dynamical system method, we obtain the exact parametric expressions of the travelling wave solutions of the Wu–Zhang system. Our approach is much different from the existing literature studies on the Wu–Zhang system. Moreover, we also study the fractional derivative of the Wu–Zhang system. Finally, by comparison between the integer-order Wu–Zhang system and the fractional-order Wu–Zhang system, we see that the phase portrait, nonzero equilibrium points, and the corresponding exact travelling wave solutions all depend on the derivative order α. Phase portraits and simulations are given to show the validity of the obtained solutions.

Dynamical system analysis for a scalar–tensor model with Gauss–Bonnet coupling

In this paper, we study the dynamics of a scalar–tensor model of dark energy in which a scalar field that plays the role of dark energy, non-minimally coupled to the Gauss–Bonnet invariant in four dimensions. We utilize the dynamical system method to extract the critical points of the model and to conclude about their stability, we investigate the sign of the corresponding eigenvalues of the perturbation matrix at each point numerically. For exponential form of the scalar field potential and coupling function, we find five stable points among the critical points of the autonomous system. We also find four scaling attractor solutions with the property that the ratio of dark energy to dark matter density parameters are of order one. These solutions give the hope to alleviate the well-known coincidence problem in cosmology.