Qualitative Theory of Dynamical Systems最新文献

In this paper, we study the following nonlocal problem in (mathbb R^3)

where (lambda >0) is a real parameter and (mu >0) is small enough. Under some suitable assumptions on V(x) and f(x, u), we prove the existence of ground state solutions for the problem when (lambda ) is large enough via variational methods. In addition, the concentration behavior of these ground state solutions is also investigated as (lambda rightarrow +infty ).

Abstract

Distributed-order calculus can summarize the intrinsic multiscale effects of integer and fractional order operators, and construct a more complex physical model. The paper is devoted to study the time distributed-order wave equation. First, we give the definition of distributed-order integral operators (I ^{(mu )}) in (alpha in [1,2]) , and from the definition of the integral operator, we found that the operator has similar properties to the fractional integral operators. Next, according to the properties of the distributed-order integral operator and Laplace transform, we obtain the expression of the solution of the distributed-order wave equation. Then we use the resolvent operator to estimate the solution operators. At last, we further studied the liner or semilinear wave problem with the distributed-order derivative on (mathbb {R}^N) and used the contraction mapping principle to prove the existence and uniqueness of mild solution.

Motivation/Development: In order to investigate the shallow-water waves, researchers have introduced many nice models, e.g., a Boussinesq-Burgers system for cetain shallow-water waves near an ocean beach/inside a lake, which we study here via computerized symbolic computation. Originality/Novelty with Potential Application: Concerning the height deviating from the equilibrium position of water as well as the field of horizontal velocity, we now construct a hetero-Bäcklund transformation coupling that system to a known partial differential system, as well as two sets of the similarity reductions, starting at that system towards a known ordinary differential equation. Both our hetero-Bäcklund transformation and similarity reductions lean upon the dispersive power in the shallow water. Results could help the further study on the oceanic/laky shallow-water dynamics.

Abstract

This paper is devoted to the proof of almost global existence results for the d-dimensional beam equation with derivative nonlinear perturbation by using Birkhoff normal form technique and the so-called tame property in a Gevrey space.

This paper discusses the approximate controllability of Hilfer fractional stochastic differential system involving non-instantaneous impulses with Rosenblatt process and Poisson jumps. By utilising stochastic analysis, semigroup theory, fractional calculus, and Krasnoselskii’s fixed point theorem, we prove our primary outcomes. Firstly, we prove the approximate controllability of the Hilfer fractional system. As a final step, we provide an example to highlight our discussion.

In this article, an inequality which contains bound of Csiszár divergence is generalised via diamond integral on time scales by utilizing the Hermite polynomial. Various constraints of Hermite polynomial are employed to provide some improvements of this new inequality. Bounds of different divergence measures are obtained by using particular convex functions. Furthermore, in seek of applications in mathematical statistics, bounds of different divergence measures are estimated on diverse fixed time scales. The paper addresses new results which are generalized (unified) form of both discrete and continuous results in literature (As time scales calculus unifies both discrete and continuous cases). Moreover, diamond integral can be used to study hybrid discrete-continuous systems.

The occurrence of cannibalism is common in natural colonies and can substantially affect the functional relationships between predators and prey. Despite the belief that cannibalism stabilizes or destabilizes predator–prey models, its effects on prey populations are not well-understood. In this study, we propose a discrete-time prey–predator model to examine the presence and local stability of biologically possible equilibria. We employ the center manifold theorem and normal theory to investigate the various types of bifurcations that arise in the system. The findings of our study reveal that the model exhibits transcritical bifurcation at its trivial equilibrium. In addition, the discrete-time predator–prey system demonstrates period-doubling bifurcation in the vicinity of both its boundary equilibrium and interior equilibrium. Furthermore, we analyze the existence of Neimark–Sacker bifurcation around the interior equilibrium point. We demonstrate that cannibalism in the prey population can lead to periodic outbreaks, but these outbreaks are limited to the prey population and do not affect predation. In order to regulate the periodic oscillations and other bifurcating and fluctuating behaviors of the system, various chaos control strategies are executed. Additionally, extensive numerical simulations are carried out to validate and substantiate the analytical findings. We utilized the software Mathematica 12.3, which is an efficient and effective computing tool that enables symbolic and numerical computations to carry out numerical simulations.

In this paper we provide a complete characterization of a class of unbounded asymmetric stationary solutions of the lattice Nagumo equations. We show that for any bistable cubic nonlinearity and arbitrary diffusion rate there exists a two-parametric set of equivalence classes of generally asymmetric stationary solutions which diverge to infinity. Our main tool is an iterative mirroring technique which could be applicable to other problems related to lattice equations. Finally, we generalize the result for a broad class of reaction functions.

In this paper, the main work is to study the N-soliton solutions for a higher-order coupled nonlinear Schrödinger system by using the method of Riemann–Hilbert. In the process of research, starting with the spectral analysis for the x-part of the Lax pair, we formulate the Riemann–Hilbert problem for the higher-order coupled nonlinear Schrödinger system. Then we infer the symmetric relation of the potential matrix and scattering data, from which we can find the zero structure of the Riemann–Hilbert problem. Moreover, we can obtain the unified formulas of the N-soliton solutions for the higher-order coupled nonlinear Schrödinger system by solving the non-regular Riemann–Hilbert problem. In addition, the dynamical behaviors of the single-soliton solution, the two-soliton solutions and the three-soliton solutions are analyzed by choosing appropriate parameters.

In this paper, we investigate the existence and uniqueness of mild solutions to the fractional Navier–Stokes equations related to time derivative of order (alpha in (0,1)). And the mild solution is associated with the sublaplacian provided by the left invariant vector fields on the Heisenberg group. We demonstrate that when the nonlinear external force term matches the applicable conditions, the global mild solution can be obtained by using improved Ascoli–Arzela theorem and Schaefer’s fixed point theorem.