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.

Fractals have been studied in areas such as mathematics, physics, chemistry, social sciences, computing, economics and biology. Fractal networks have many interesting properties, such as recursive selfsimilarity, that are present in many real networks. In this paper we study the geometrical and topological properties of fractal networks (Sierpiński triangle, the Sierpiński carpet and the Koch snowflake). Furthermore, we establish relationships, in the fractal structures studied, between the geometric properties associated with hyperbolicity and the topological indices.

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.

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.

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.

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

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.

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.

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.

In this manuscript new oscillation criteria are given for solutions of h-order delay difference equation in the form ∆ E n + p n E n − α = 0 . Here, four cases of p are considered; for α∈N , as oscillatory periodic sequences with periods 2 α,(2 α+1) and ( α+1) , and a quickly oscillating sequence such that p n p n + 1 < 0 . It is well known that coefficient functions has a very important role in the behavior of the solutions of an equation. The coefficients sequence p has been considered as positive sequences in all oscillation criteria so far. But It is considered as a oscillatory periodic sequences and quickly oscillatory sequences for the first time in this study. We use a new technique to prove our results. Also some examples are provided to demonstrate the effectiveness of the main results.

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→∞.

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.

We offer a detailed treatment of minimal, maximal, dissipative, accumulative, self‐adjoint operator realizations with exit from space of boundary–value problem for Sturm‐Liouville equation with unbounded operator coefficient having discrete spectrum andwith boundary condition dependent on Herglotz‐Nevanlinna function of eigenparameter. We also study self‐adjoint operator realizations having purely discrete spectrum or having continuous spectrum which coincide with any interval on real axis. That is done in terms of boundary conditions. In addition, in particular case we obtain asymptotics of spectrum of self‐adjoint operators with discrete spectrum.

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.

Large constant-sign solutions for boundray value problems (Bvps) of the partial discrete difference equation involving ( p,q)-Laplacian is investigated in this paper. The method is based on the critical point theory and truncation techniques. Even reducing to the ordinary difference equations, our results generalize and significantly improve some existing ones. In the end, the obtained results are exemplified based on two examples.

We establish necessary and sufficient conditions for a weight v guaranteeing the boundedness of multilinear Riemann-Liouville operators from ∏ j = 1 m L p j to L v q .

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.