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In this paper, two methods are applied to solve the regularized Schamel equation. Firstly, by using the singular planar dynamical system method, we discover its peakon structure which was not reported before. We also derive some new explicit traveling wave solutions of this equation, including various solitary wave solutions, periodic wave solutions and compactons. Especially, for the first time, we find the W-shape solitary wave solutions and W-shape periodic wave solutions of the equation. Then, in order to discover more wave phenomena, a deep learning framework is introduced to solve complicated initial boundary value problems of this equation. In comparison with the exact solutions given previously, our deep learning framework is reliable and highly accurate in capturing the dynamical behavior of the traveling wave solutions of the equation. As an application, we use it to solve a specific initial boundary value problem and obtain a new data-driven solution. [ABSTRACT FROM AUTHOR]

The classic linear Mathieu equation is one of the archetypical differential equations which has been studied frequently by employing different analytical and numerical methods. The Mathieu equation with cubic nonlinear term, also known as Mathieu–Duffing equation, is one of the many extensions of the classic Mathieu equation. Nonlinear characteristics of such equation have been investigated in many papers. Specifically, the method of multiple scale has been used to demonstrate the pitchfork bifurcation associated with stability change around the first unstable tongue and Lie transform has been used to demonstrate the subharmonic bifurcation for relatively small values of the undamped natural frequency. In these works, the resulting bifurcation diagram is represented in the parameter space of the undamped natural frequency where a constant value is allocated to the parametric frequency. Alternatively, this paper demonstrates how the Poincaré–Lindstedt method can be used to formulate pitchfork bifurcation around the first unstable tongue. Further, it is shown how higher order terms can be included in the perturbation analysis to formulate pitchfork bifurcation around the second tongue, and also subharmonic bifurcations for relatively high values of parametric frequency. This approach enables us to demonstrate the resulting global bifurcation diagram in the parameter space of parametric frequency, which is beneficial in the bifurcation analysis of systems with constant undamped natural frequency, when the frequency of the parametric force can vary. Finally, the analytical approximations are verified by employing the numerical integration along with Poincaré map and phase portraits. [ABSTRACT FROM AUTHOR]

In this paper, we propose a reaction–diffusion mathematical model augmented with self/cross-diffusion in 2D domain which describes the oncolytic virotherapy treatment of a tumor with its growth following the logistic law. The tumor cells are divided into uninfected and infected cells and the virus transmission is supposed to be in a direct mode (from cell to cell). In the absence of cross-diffusion, we establish well posedness of the problem, non-negativity and boundedness of solutions, nonexistence of positive solutions, local and global stability of the nontrivial steady-state and the nonoccurrence of Turing instability. In the presence of cross-diffusion, we prove the occurrence of Turing instability by using the cross-diffusion coefficient of infected cells as a parameter. To have an idea about different patterns, we derive the corresponding amplitude equation by using the nonlinear analysis theory. In the end, we perform some numerical simulations to illustrate the obtained theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, a perturbed quintic BBM equation with weak backward diffusion and dissipation effects is investigated. By applying geometric singular perturbation theory and analyzing the perturbations of a Hamiltonian system with a hyper-elliptic Hamiltonian of degree six, we prove the existence of isolated periodic wave solutions with certain wave speed in an open interval. It is also shown that isolated periodic wave solutions persist for any energy parameter h in an open interval under small perturbation. Furthermore, we prove that the wave speed c (h) of periodic wave is strictly monotonically increasing with respect to h by analyzing Abelian integral having three generating elements. Moreover, the upper and lower bounds of the limiting wave speed are obtained. Our analysis is mainly based on Melnikov theory, Chebyshev criteria, and symbolic computation, which may be useful for other problems. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the number of nontrivial limit cycles in a kind of piecewise smooth generalized Abel equation d x d t = x p p − 1 + ∑ i = 1 2 i (A i − (t) x p + B i − (t) x 2 p − 1) , − 1 ≤ t ≤ 0 , x ∈ R , − x p p − 1 + ∑ i = 1 2 i (A i + (t) x p + B i + (t) x 2 p − 1) , 0 ≤ t ≤ 1 , x ∈ R , where p ∈ Z + ∖ { 1 } , A 1 ± = a 1 0 ± + a 1 1 ± t , B 1 ± = b 1 0 ± + b 1 1 ± t , A 2 ± = a 2 0 ± + a 2 1 ± t , B 2 ± = b 2 0 ± + b 2 1 ± t and | | ≪ 1. Under the condition b 1 1 + ≠ 0 , employing Melnikov functions of any order and using properties of Chebyshev systems, we prove that if p is odd, then the maximum number of nontrivial limit cycles bifurcating from the periodic annulus of the unperturbed system is 6 and it is attainable, and if p is even, then the maximum number is 3, and it can be attained too. [ABSTRACT FROM AUTHOR]

In this paper, the methodologies of dynamical systems and singular traveling wave theory developed by [Li & Chen, 2007] are applied to find the solutions in the form of u (x , t) = ϕ (x − c t) e i (κ x − ω t) for the Hirota-type peakon equation posed by [Anco & Mobasheramini, 2017; Anco et al., 2021]. For the function ϕ (ξ) therein, under the parameter conditions of κ = 5 3 , c = 8 0 7 ω 1 0 0 or c ≠ 8 0 7 ω 1 0 0 , the existence of some possible bounded solutions (solitary wave solutions, periodic wave solutions and periodic peakons, as well as compactons) is proved, with exact explicit parametric representations obtained, except for the peakon solution. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the bifurcations and exact traveling wave solutions of the Lakshmanan–Porsezian–Daniel model with parabolic law nonlinearity. Employing the bifurcation theory of planar dynamical system, we obtain the phase portraits of the corresponding traveling wave system. The existence of the two singular straight lines leads to the dynamical behavior of solutions with two scales. Corresponding to some special level curves, we give the explicit exact solutions under different parameter conditions, containing peakon solutions, solitary wave solutions, kink and anti-kink wave solutions, periodic wave solutions and periodic peakon solutions. Moreover, we collect all these traveling solutions into a theorem. Finally, we discuss a small perturbation of the LPD equation and persistence of the traveling waves. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to form a complete group classification for the Gilson–Pickering equation. Exact traveling wave solutions and bifurcations of the Gilson–Pickering equation are studied by the method of dynamical systems. The study on the wave solutions of the model derives a planar Hamiltonian system. Based on phase portraits, many exact explicit parametric representations of wave solutions are obtained under different parametric conditions. [ABSTRACT FROM AUTHOR]

This paper studies traveling wave solutions of a generalized Sasa–Satsuma equation introduced in [Adem et al., 2020]. For the cases of n = 0 , 1 , 2 , under given parameter conditions, the bifurcations of traveling wave solutions in the parameter space are investigated for the corresponding traveling systems. All possible explicit exact parametric representations of various solutions are obtained. [ABSTRACT FROM AUTHOR]

A modified generalized KdV equation is considered in this paper. Under the given parameter conditions, the corresponding traveling wave system is a singular planar dynamical system with three singular straight lines. The bifurcations and traveling wave solutions of the system are investigated in the parameter space from the perspective of dynamical systems. The existence of solitary wave solutions, periodic peakon solutions, pseudo-peakon solutions, kink and anti-kink wave solutions and compactons is proved. Furthermore, possible exact explicit parametric representations of various solutions are given. Particularly, the model has uncountably infinite many solitary wave and pseudo-peakon solutions. [ABSTRACT FROM AUTHOR]

The article considers the question of local dynamics of the logistic equation with delays and the delay coefficients being the nonlinear functions. It is supposed that one of the parameters, characterizing the delay value, is sufficiently large. The stability criterion is formulated and critical cases in the problem of balance state stability are defined. The main content of the paper is focused on studying the local dynamics in cases close to critical. In nature, these critical cases are infinite dimensional. Special nonlinear boundary-value problems of the parabolic type are constructed, which play the role of normal forms. Their nonlocal dynamics defines the behavior of the solutions from the small neighborhood of the balance state of the initial equation. The conclusions on the role of nonlinearity in the delay coefficients, in the behavior of the dynamic properties of the solutions are summarized. [ABSTRACT FROM AUTHOR]