In this paper, we investigate the long-term stability of classical solution to 3D one-fluid Euler-Poisson system of electrons with non-zero vorticity. It is shown that the classical solution is well-posed for a lifesapn Tδ at least εδ, where ε>0 is the size of the initial perturbed density and velocity, δ>0 is the size of the initial vorticity. This implies that the lifespan Tδ only essentially depends on the size δ of the initial vorticity.

In this paper, we study the highly dispersive nonlinear perturbation Schrödinger equation, which has arbitrary form of Kudryashov’s with sextic-power law refractive index and generalized non-local laws. For the equation has highly dispersive nonlinear terms and higher order derivatives, it cannot be integrated directly, so we build an integrable factor equation for the approximated equation and apply the trial equation method and the complete discrimination system for polynomial method to create new soliton solutions. On the other hand, we use the bifurcation theory to qualitatively analyze the equation and find the model has periodic solutions, bell-shaped soliton solutions and solitary wave solutions via phase diagrams. The topological stability of the solutions with respect to the parameters is explored in order to better understand the effect of parameters perturbations on the stability of the model’s solutions. Furthermore, we analyze the modulation instability and give the corresponding linear criterion. After accounting for external perturbation terms, we analyze the chaotic behaviors of the equation through the largest Lyapunov exponents and phase diagrams.

The aim of this paper is to derive many novel formulas involving the sum of powers of consecutive integers, the Bernoulli polynomials, the Stirling numbers and moments arise from conditional probability, moment generating functions and arithmetic functions by using the methods and techniques, which are used in discrete distributions in statistics such as uniform distribution, moment generating functions, and other probability distributions. Moreover, relations among the generalized Euler totient function, finite distributions containing special numbers and polynomials, discrete probability formula, and other special function are given. By using the Riemann zeta function and the Liouville's function, we derive a novel moment formula probability distribution on the set positive integers. Finally, by using approximation formulas for certain family of finite sums, approximation formulas for the sum of powers of consecutive integers involving the Bernoulli polynomials,and certain classes conditional probability involving the Laplace's rule of succession are derived.

In this work, we study and solve the normalized Ricci flow equation for circle bundles over surfaces. Moreover, we study the asymptotic behavior of the solutions and their connections to some model geometries.

We consider a degenerate/singular wave equation in one dimension, with drift and in presence of a leading operator which is not in divergence form. We impose a homogeneous Dirichlet boundary condition where the degeneracy occurs and a boundary damping at the other endpoint. We provide some conditions for the uniform exponential decay of solutions for the associated Cauchy problem.

Consider the nonlinear difference equations of the form L u = f m ( u ) , m ∈ Z , where L is a Jacobi operator given by L u m = a m u m + 1 + a m − 1 u m − 1 + b m u m for m∈Z, { a m } and { b m } are real valued T-periodic sequences, and f:Z×R→R. Applying critical point theory and a new analytical method, we obtain that the above problem has ground state solutions and infinitely many geometrically distinct solutions under the local superlinear condition lim | x | → ∞ ∫ 0 x f m ( t ) dt | x | 2 = ∞ uniformly in m∈ D for some set D⊂Z instead of the global superlinear condition lim | x | → ∞ ∫ 0 x f m ( t ) dt | x | 2 = ∞ uniformly in m∈Z.

In this paper, we consider the following 2D incompressible chemotaxis-Navier-Stokes equations with the fractional diffusion { ∂ t n + u · ∇ n − ∆ n = − ∇ · ( n ∇ c )+ n ( 1 − n )( n − a ) , ∂ t c + u · ∇ c − ∆ c = − cn , ∂ t u + u · ∇ u + ∧ 2 α u + ∇ P = − n ∇ ϕ , where ∧ : = ( − ∆ ) 1 2 and α ∈ [ 1 2 , 1 ] . By establishing the new priori estimates, we get the global well-posedness for the above system with large initial data.

This research deals with the Kaup-Newell model (KNM), a class of nonlinear Schrödinger equations with important applications in plasma physics and nonlinear optics. Soliton solutions are essential for analyzing nonlinear wave behaviors in different physical systems, and the KNM is also significant in this context. The model’s ability to represent sub-picosecond pulses makes it a significant tool for the research of nonlinear optics and plasma physics. Overall, the KNM is an important research domain in these areas, with ongoing efforts focused on understanding its various solutions and potential applications. A new version of the generalized exponential rational function method (nGERFM) and ( G ′ G 2 ) -expansion function methods are utilized to discover diverse soliton solutions. The nGERFM facilitates the generation of multiple solution types, including singular, shock, singular periodic, exponential, combo trigonometric, and hyperbolic solutions in mixed forms. Thanks to ( G ′ G 2 ) -expansion function method, we obtain trigonometric, hyperbolic, and rational solutions. The modulation instability (MI) of the proposed model is examined, with numerical simulations complementing the analytical results to provide a better understanding of the solutions’ dynamic behavior. These results offer a foundation for future research, making the solutions effective, manageable, and reliable for tackling complex nonlinear problems. The methodologies used in this study are robust, influential, and practicable for diverse nonlinear partial differential equations (NLPDEs); to our knowledge, for this equation, these methods of investigation have not been explored before. The accuracy of each solution has been verified using the Maple software program.

This paper focuses on the three-dimensional(3D) incompressible anisotropic Boussinesq system while the velocity of fluid only involves horizontal dissipation and the temperature has a damping term. By utilizing the structure of the system, the energy methods and the means of bootstrapping argument, we prove the global stability property in the Sobolev space H k ( R 3 ) ( k ≥ 3 ) of perturbations near the hydrostatic equilibrium. Moreover, we take an effective approach to obtain the optimal decay rates for the global solution itself as well as its derivatives. In this paper, we aim to reveal the mechanism of how the temperature helps stabilize the fluid. Additionally, exploring the stability of perturbations near hydrostatic equilibrium may provide valuable insights into specific severe weather phenomena.

Linear growth of spherically symmetric solutions to Navier-Stokes equations in R N with degenerate viscosity and free boundary

Based on partial Fourier series analysis, we adapt on a model case a new approach to classical results obtained in the literature describing the singularities of a family a solutions of a second order elliptic problems on open non-convex planar sectors. The method allows the exhibition of singular and regular frequencies, explicit decomposition and description of coefficients of singularities of the solution. As a main result, explicit and sharp estimates with respect to the opening angle parameter are obtained via this method. They are not uniform near π where corners have opening angle generating a jump of singularity in Sobolev exponent, contrarily to the results obtained in A. Tami (2016),(2019),(2021) for harmonic and/or biharmonic problems on a family of convex planar sectors.

Cauchy matrix approach for the discrete Ablowitz-Kaup-Newell-Segur equations is reconsidered, where two ‘proper’ discrete Ablowitz-Kaup-Newell-Segur equations and two ‘unproper’ discrete Ablowitz-Kaup-Newell-Segur equations are derived. The ‘proper’ equations admit local reduction, while the ‘unproper’ equations admit nonlocal reduction. By imposing the local and nonlocal complex reductions on the obtained discrete Ablowitz-Kaup-Newell-Segur equations, two local and nonlocal discrete complex modified Korteweg-de Vries equations are constructed. For the obtained local and nonlocal discrete complex modified Korteweg-de Vries equations, soliton solutions and Jordan-block solutions are presented by solving the determining equation set. The dynamical behaviors of 1-soliton solution are analyzed and illustrated. Continuum limits of the resulting local and nonlocal discrete complex modified Korteweg-de Vries equations are discussed.

Two types of short-time Fourier transforms involving respectively nonlinear modulation and nonlinear translation are designed in this note. Some important properties such as twin in time-frequency domain and orthogonality relations are investigated. Moreover, Lieb type inequalities and uncertainty principle are established.

A document by Aykut Toplama. Click on the document to view its contents.

We consider conditions for the decay in time of solutions of non-homogenous hyperbolic equations. It is proven that solutions of the equations go to 0 in L 2 at infinity if and only if an equation’s right-hand side uniquely determines the initial conditions in a certain way. We also obtain that a hyperbolic equation has a unique solution that fades when t→∞.

We develop the Riemann-Hilbert (RH) method for the generalized inhomogeneous Hirota equation with zero boundary condition. The RH problem is related to two kinds of scattering data: N simple poles and one N-order pole. Here we consider that when the scattering data have one or more higher-order poles, the formulas of bound-state (BS) solitons and multiple BS solitons are obtained, and the interaction between solitons and BS solitons are shown.

The uniform exponential stabilities (UESs) of two hybrid control systems comprised of a wave equation and a second-order ordinary differential equation are investigated in this study. Linear feedback law and local viscosity are considered, as are nonlinear feedback law and internal anti-damping. The hybrid system is first reduced to a first order port-Hamiltonian system with dynamical boundary conditions, and the resulting system is discretized using the average central-difference scheme. Second, the UES of the discrete system is obtained without prior knowledge of the exponential stability of the continuous system. The frequency domain characterization of UES for a family of contractive semigroups and the discrete multiplier approach are used to validate the main conclusions. Finally, the Trotter-Kato Theorem is used to perform a convergence study on the numerical approximation approach. Most notably, the exponential stability of the continuous system is derived by the convergence of energy and UES, which is a novel approach to studying the exponential stability of some complex systems. Numerical simulation is used to validate the effectiveness of the numerical approximating strategy.

This paper focuses on the existence of normalized solutions for the Chern-Simons-Schrödinger system with mixed dispersion and critical exponential growth. These solutions correspond to critical points of the underlying energy functional under the L 2 -norm constraint, namely, ∫ R 2 u 2 d x = c > 0 . Under certain mild assumptions, we establish the existence of nontrivial solutions by developing new mathematical strategies and analytical techniques for the given system. These results extend and improve the results in the existing literature.