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In this paper, we use the techniques from dynamical systems and singular traveling wave theory developed by [Li & Chen, 2007] to investigate the exact explicit solutions for the perturbed Gerdjikov–Ivanov (GI) equation. By considering the corresponding dynamical system and finding the bifurcations of phase portraits for the amplitude component of the traveling solutions, the dynamical behavior can be revealed. Under different parameter conditions, exact explicit solutions of the perturbed GI equation are found.

Due to the essential difficulty of establishing an appropriate variational framework on a suitable working space, how to apply the critical point theory for showing the existence and multiplicity of periodic solutions of continuous-time difference equations remains a completely open problem. New ideas including gluing arguments are introduced in this work to overcome such a difficulty. This enables us to employ the critical point theory to construct uncountably many periodic solutions for a class of superlinear continuous-time difference equations without assuming symmetry properties on the nonlinear terms. The obtained solutions are piecewise differentiable in some cases, distinguishing continuous-time difference equations from ordinary differential equations qualitatively. To the best of our knowledge, this is the first time in the literature that the critical point theory has been used for such types of problems. Our work may open an avenue for studying discrete nonlinear systems with continuous time via the critical point theory.

Seasonality is repetitive in the ecological, biological and human systems. Seasonal factors affect both pathogen survival in the environment and host behavior. In this study, we considered a five-dimensional system of ordinary differential equations modeling an epidemic in a seasonal environment with a general incidence rate. We started by studying the autonomous system by investigating the global stability of steady states. Later, we proved the existence, uniqueness, positivity and boundedness of a periodic orbit in a non-autonomous system. We demonstrate that the global dynamics are determined using the basic reproduction number ${{ℛ}}_0$ which is defined by the spectral radius of a linear integral operator. We showed that if ${{ℛ}}_0<1$, then the disease-free periodic solution is globally asymptotically stable and if ${{ℛ}}_0>1$, then the trajectories converge to a limit cycle reflecting the persistence of the disease. Finally, we present a numerical investigation that support our results.

In this paper, a population system with cross-diffusion and habitat complexity is selected as study object. We investigate that how cross-diffusion and habitat complexity destabilize the otherwise stable periodic solutions of the ODEs to generate the new abundant spatial Turing patterns. By utilizing the local Hopf bifurcation theorem and perturbation theory, we establish a formula to determine the Turing instability of periodic solutions of the population system with cross-diffusion and habitat complexity. Finally, numerical simulations are performed to verify theoretical analysis, simultaneously, we verify the formation process of spatial Turing patterns when the cross-diffusion coefficients and habitat complexity change.

In this paper, a mathematical model for a solid spherically symmetric vascular tumor growth with nutrient periodic supply and time delays is studied. Compared to the apoptosis process of tumor cells, there is a time delay in the process of tumor cell division. The cells inside the tumor obtain nutrient $\sigma (r,t)$ through blood vessels, and the tumor attracts blood vessels at a rate proportional to $\alpha (t)$. So, the boundary value condition

Considering the food diversity of natural enemy species and the habitat complexity of prey populations, a pest-natural enemy model with non-monotonic functional response is proposed for biological management of Bemisia tabaci. The dynamic characteristics of the proposed model are analyzed. In addition, considering that the conversion from prey to predator has a certain time lag rather than instantaneous, a time delay is introduced into this model, and it is shown that the Hopf bifurcation occurs at the interior equilibrium when the time delay is used as the bifurcation parameter. Furthermore, the values of the parameters that determine the direction of the Hopf bifurcation as well as the stability of the periodic solution are calculated. In order to illustrate the theoretical analysis results, numerical simulations and validation are carried out to demonstrate the effects of non-monotonic functional response, additional food supply and habitat complexity.

Based on the inequality analysis, matrix theory and spectral theory, a class of general periodic neural networks with delays and impulses is studied. Some sufficient conditions are established for the existence and globally exponential stability of a unique periodic solution. Furthermore, the results are applied to some typical impulsive neural network systems as special cases, with a real-life example to show feasibility of our results.

In this paper, a discrete version of continuous non-autonomous predator–prey model with delays is investigated. By using Gaines and Mawhin's continuation theorem of coincidence degree theory and the method of Lyapunov function, some sufficient conditions for the existence and globally asymptotically stability of positive periodic solution of difference equations in consideration are established. Finally, some numerical examples are given to verify the theoretical analysis.

A modification of the homotopy perturbation method is proposed by taking advantage of the enhanced perturbation method and the parameter expanding technology. A generalized oscillatory equation and some nonlinear oscillators as the special cases of this equation are considered as examples to outline the basic properties of the modification, and the result is of high accuracy.

Under investigation in this paper is the long-wave-short-wave resonance system, which can describe a variety of nonlinear wave phenomena such as the two-dimensional packets of capillary-gravity waves in hydrodynamics and the optical-terahertz waves. The intended aim is carried out via considering a traveling wave reduction, adopting a modified version of the Jacobi elliptic expansion method and employing the Weierstrass elliptic function method to derive such analytic solutions as the bright and dark soliton solutions, periodic solutions, trigonometric-function and elliptic-function solutions in fluid mechanics.

Using the Darboux transformation method, the general Lax equation is solved and a collection of new exact solutions together with one-soliton solutions, singular one-soliton solutions, periodic solutions, singular periodic solution, two-soliton solutions, singular two-soliton solutions, two-periodic solutions and singular two-periodic solutions is obtained. Using traveling wave transformation, the Lax equation is transfigured to a conservative dynamical system (CDS) of dimension four with three equilibrium points involving two parameters $\gamma$ and v. The CDS has various quasi-periodic motions for fixed values of the parameters $\gamma$ and v at different initial conditions. Furthermore, effects of the parameters $\gamma$ and v are shown on the quasiperiodic motions of the CDS by means of phase sections and time series plots. This approach can be applied to a heterogeneity of nonlinear model equations or partial differential equations for describing their inherent nonlinear phenomena.

Abundant hybrid solitary wave solutions have been investigated for the $(3+1)$-dimensional nonlinear evolution equation with main part mKdV equation (NLEE-mKdV) in its first study via the Lie symmetry method. The latter has been harnessed to attain 10-dimensional vector fields of symmetries from the converted NLEE-mKdV to a simpler equation under special transformation. From the derived symmetries, the corresponding one-dimensional optimal system has been constructed. Furthermore, the converted NLEE-mKdV is reduced via five subalgebras of symmetries to obtain nine novel solutions, which are shifted by the considered transformation to reach NLEE-mKdV’ solutions that possess different dynamics depending on a special ansatz in adjusting the arbitrary functions and parameters. Therefore, many significant solitary wave patterns are achieved. For example, dark, bright, periodic, dipole, damped periodic, breather, kink, and their interactions are well depicted in 3D and contour plots. Most importantly, a fascinating solitary wave solution is first explored; the intrinsic insight appears on the periodically-parabolic-periodic background, which is collided with a bright soliton solution to induce a parabolic-humps breather on its top.

The propagations are generally described through nonlinear Schrödinger equation (NLSE) in the optical solitons. In the NLSEs, the higher order NLSE with derivative non-Kerr nonlinear terms is a model that depicts propagation of pulses beyond ultra-short range in optical communication system. Several novel exact solutions of different kinds such as solitons, solitary waves and Jacobi elliptic function solutions are achieved via using modified extended mapping technique. Different kinds of exact results have prestigious exertions in engineering and physics. Structures of solitons different kinds are shown graphically by giving suitable values to parameters. The physical interpretations of solutions can be understand through structures. Several exact solutions and computing work confirm the supremacy and usefulness of the current technique.

The $sinh$-Gordon model is an important model in special nonlinear partial differential equations (PDEs) which is arising in solid-state physics, mathematical physics, fluid dynamics, fluid flow, differential geometry, quantum theory, etc. The exact solutions in the type of solitary wave and elliptic functions solutions are created of $sinh$-Gordon model by employing modified direct algebraic scheme. Moments of a few solutions are also depicted graphically. These solutions helps the physicians and mathematicians to understand the physical phenomena of this model. This technique can be utilized on other models to launch further exclusively novel solutions for other categories of nonlinear PDEs occurring in mathematical Physics.

We present lump-type solutions and interaction solutions to an extended (3+1)-dimensional Jimbo–Miwa-like equation. Three classes of lump-type solutions are obtained by the Hirota bilinear method. Interaction solutions are among lump-type solutions, two kink waves and periodic waves, and between two kink waves and a periodic wave are computed. Dynamical characters of the obtained solutions are graphically exhibited. These wave solutions enrich the dynamical theory of higher-dimensional nonlinear dispersive wave equations.

In this paper, a variable-coefficient KdV equation in a fluid, plasma, anharmonic crystal, blood vessel, circulatory system, shallow-water tunnel, lake or relaxation inhomogeneous medium is discussed. We construct the reduction from the original equation to another variable-coefficient KdV equation, and then get the rational, periodic and mixed solutions of the original equation under certain constraint. For the original equation, we obtain that (i) the dispersive coefficient affects the solitonic background, velocity and amplitude; (ii) the perturbed coefficient affects the solitonic velocity, amplitude and background; (iii) the dissipative coefficient affects the solitonic background, and there are different mixed solutions under the same constraint with the dispersive, perturbed and dissipative coefficients changing.

In this paper, we study exact solutions of the (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation. We employ the Hirota bilinear method to obtain the multi-solitary wave solutions, soliton resonant solutions, periodic solutions and interactional solutions and periodic resonant solutions. The corresponding asymptotic features and images are also clearly given.

Memristor, as the future of artificial intelligence, has been widely used in pattern recognition or signal processing from sensor arrays. Memristor-based recurrent neural network (MRNN) is an ideal model to mimic the functionalities of the human brain due to the physical properties of memristor. In this paper, the periodicity for memristor-based Cohen–Grossberg neural networks (MCGNNs) is studied. The neural network (NN) considered in this paper is based on the memristor and involves time-varying delays, distributed delays and impulsive effects. The boundedness and monotonicity of the activation function are not assumed. By some inequality technique and contraction mapping principle, we prove the existence, uniqueness and exponential stability of periodic solution for MCGNNs. Finally, some numeral examples and comparisons are provided to illustrate the validation of our results.

In this paper, we develop Kaplan–Yorke's method and consider the existence of 2r/(2k+1)-periodic solutions for certain three-dimensional delay differential systems. We also study Hopf bifurcations of this kind of periodic solutions for the system with a parameter and present some application examples of our main results.

A system that consists of two impacting oscillators with damping has been considered in this paper. In the first part, a method of analytical determination of the existence of periodic solutions to the equations of motion and a method of analysis of the stability of these solutions are presented. The results of the computations carried out by these methods have been illustrated with a few examples. In the second part of the paper, the results of some numerical investigations are presented. The goal of these studies is to determine, in which regions of parameters characterizing the system, the periodic motion with one impact per period exists and is stable.