Qualitative Theory of Dynamical Systems最新文献

In this paper, we consider the following Schrödinger equation:

where ( N ge 3 ), ( a>0 ), (sigma >0), and ( lambda in mathbb {R}) appears as a Lagrange multiplier. Assume that the nonlinear term f satisfies conditions only in a neighborhood of zero. For f has a subcritical growth, we prove the existence of the positive normalized solution for the equation with sufficiently small (sigma >0). For f has a supercritical growth, we derive the existence of the positive normalized solution for the equation with (sigma >0) large enough. In addition, we also obtain infinitely many normalized solutions with sufficiently small (sigma >0) for the subcritical case.

In this paper, we study an SICA model with a standard incidence rate, where the contact rate (beta ) is controlled by the Ornstein–Uhlenbeck process. We first prove the existence and uniqueness of the global positive solution, and by constructing an appropriate Lyapunov function, we demonstrate that when (R_0^s > 1), the system has a stationary distribution. Furthermore, we obtain a concrete expression of the probability density function near the quasi-positive equilibrium point. By constructing another suitable Lyapunov function, we also derive a threshold value (R_0^e) for disease extinction, and when (R_0^e < 1), the disease extinguishes at an exponential rate. Finally, our conclusions are verified through numerical simulations.

This article explores two distinct issues. To begin with, we analyze the Pontriagin maximum principle concerning fractional delay differential equations. Furthermore, we investigate the most effective method for resolving the control problem associated with Eq. (1.1) and its corresponding payoff function (1.2). Subsequently, we explore the Pontryagin Maximum principle within the framework of Volterra delay integral equations (1.3). We enhance the outcomes of our investigation by presenting illustrative examples towards the conclusion of the article.

We give a complete description of inequivalent nontrivial local conservation laws of all orders for a natural generalization of the dissipative Westervelt equation and, in particular, show that the equation under study admits an infinite number of inequivalent nontrivial local conservation laws for the case of more than two independent variables.

In this work, we introduce and study two problems for diffusion equations with the distributed order fractional derivatives including the initial value problem and the terminal value problem. For the initial value problem, we establish some existence results and Hölder regularity for the mild solution. On the other hand, we also show the existence results and a decay estimate of the mild solution for the terminal value problems. Especially, the polynomial decay of the solutions to the terminal value problems is firstly included when the source function is equal to zero.

The Korteweg-de Vries–Calogero–Bogoyavlenskii–Schiff (KdV-CBS) equation is often used in dealing with long-wave propagation interactions, and is widely used in mathematics, physics, and engineering. This paper proposes a new extended (3+1)-dimensional KdV-CBS equation, and it’s never been studied. Additionally, we verify the integrability of the equation based on the Painlevé test. By employing Hirota’s method, a bilinear auto-Bäcklund transformation, the multiple-soliton solutions, and the soliton molecules of the equation are derived. New exact solutions of the equation are constructed utilizing the power series expansion method and ((G'/G))-expansion method. These exact solutions are also presented graphically. Finally, the conservation laws of the equation are obtained. Our results are helpful for understanding nonlinear wave phenomena.

This paper is concerned with the following HLS lower critical Choquard equation with a nonlocal perturbation

where (alpha in (0,N)), (N ge 3), (mu , c>0), (frac{N+alpha }{N}<q<frac{N+alpha +2}{N}), (lambda in {mathbb {R}}) is an unknown Lagrange multiplier and (h:{mathbb {R}}^Nrightarrow (0,infty )) is a continuous function. The novelty of this paper is that we not only investigate autonomous case but also handle nonautonomous situation for the above problem. For both cases, we prove the existence and discuss asymptotic behavior of ground state normalized solutions. Compared with the existing references, we extend the recent results obtained by Ye et al. (J Geom Anal 32:242, 2022) to the HLS lower critical case.

Consider the number of limit cycles of a family of systems with homogeneous components: ( {dot{x}}=y, {dot{y}}=-x^3+alpha x^2y+y^3. ) We show that there is an (alpha ^*<0) such that the system has exactly one limit cycle for (alpha in (alpha ^*,0),) while no limit cycle for the else region. This completes a previous result and also gives a positive answer to the second part of Gasull’s 3rd problem listed in the paper (SeMA J 78(3):233–269, 2021). To obtain this result, we mainly analyse the behavior of the heteroclinic separatrices at infinity.

This study primarily aims to investigate the application of the Lie symmetry method and conservation law theories in the analysis of mixed fractional partial differential equations where both Riemann–Liouville time-fractional and integer-order x-derivatives are present simultaneously. Specifically, the focus is on the (2+1) dimensional Kadomtsev–Petviashvili–Benjamin–Bona–Mahony equation. The fractionally modified equation is subjected to invariant analysis using the prolongation formula for mixed derivatives (partial _{t}^{alpha }(u_{x})) and (partial _{t}^{alpha }(u_{xxx})) for the first time. Through the introduction of a novel reduction method, we utilize the Lie symmetry technique to convert the (2+1) dimensional Kadomtsev–Petviashvili–Benjamin–Bona–Mahony equation into a fractional ordinary differential equation. It’s worth noting that this transformation is carried out without employing the Erdélyi–Kober fractional differential operator. Following this, we introduce a comprehensive expression for deriving conservation laws, involving the notion of nonlinear self-adjointness. Further, two different versatile techniques, the extended Kudryashov method and the Sardar subequation method have been used to extract a wide array of fresh sets of solitary wave solutions encompassing variations like kink, bright, singular kink, and periodic soliton solutions. To provide an intuitive grasp and investigate the ramifications of the fractional derivative parameter on these solitary wave solutions, we conduct a visual exploration employing both 3D and 2D plots.

In this work, we deal with a more general form of fractional Langevin equation. The equation’s nonlinearity term f is relevant to fractional integral and fractional derivative. By using the fixed point theorems, we study the existence and uniqueness of solutions of initial value problem for the nonlinear fractional Langevin equation and obtain some new results. Further, by using the technique of nonlinear functional analysis, we study the stability of Ulam-Hyers, Ulam-Hyers-Rassias and semi-Ulam-Hyers-Rassias for the initial value problem of nonlinear Langevin equation. Finally, some examples are given to show the effectiveness of theoretical results.