We derive the exact rate of pair production of oppositely charged scalar particles by a smooth potential step φ(x) ∝ tanh kz in three dimensions. As a check we recover from this the known results for an infinitely sharp step as well as for a uniform electric field.

Aim of this article is to investigate soliton solutions recently developed 3+1-dimensional Boiti–Leon–Manna–Pempinelli equation by utilizing newly derived approach namely, improved tanh(ϕ2)-expansion method. As a result, we succeed secure various types new for model including kink, periodic rational solutions. Some the has been discussed in form 2-,3-dimensional graphs and their contour plots v...

We provide a simple extension of Bolthausen's Morita type proof the replica symmetric formula [E. Bolthausen, Stat. Mech. Classical and Disordered Systems, pp. 63-93 (2018)] for Sherrington-Kirkpatrick model prove symmetry all $(\beta,h)$ that satisfy $\beta^2 E \operatorname{sech}^2(\beta\sqrt{q}Z+h) \leq 1$, where $q = E\tanh^2(\beta\sqrt{q}Z+h)$. Compared to (2018)], key argument is apply co...

In this paper, the perturbed Gerdjikov–Ivanov (GI) equation using a truncated M-fractional derivative is studied in mathematical nonlinear optics. We are explored its novel dark and other soliton solutions compared them with existing results. To obtain objective, two particular methods, modified extended $$\tanh$$ expansion method $$Exp_a$$ function method, implemented. exercise, an arrangement...

In this paper, we consider the (4+1)-dimensional fractional Fokas equation (FFE) with an M-truncated derivative. The extended tanh–coth method and Jacobi elliptic function are utilized to attain new hyperbolic, trigonometric, elliptic, rational solutions. addition, generalize some previous results. acquired solutions beneficial in analyzing definite intriguing physical phenomena because FFE is ...

Generally, the position closed-loop control of stepper motors requires high-precision models, however are complex nonlinear systems, and models difficult to obtain. In order solve this problem, a variable frequency controller based on hyperbolic tangent function is proposed. Similar proportional integral (PI) method, method does not require an exact model plant. The experimental results show th...

In this paper, the two-dimensional generalized nonlinear Schrödinger equations are introduced with Lax pair. The existence of pair defines integrability for partial differential equation, so integrable. Related to development was understanding that certain coherent structures called solitons play a basic role in phenomena as fluid mechanics, optics relativity, and lattice dynamics. Via Hirota b...

It is important to seek for more explicit exact solutions of nonlinear partial differential equations (NLPDEs) in mathematical physics. With the help of symbolic computation software like Maple or Mathematica, much work has been focused on the various extensions and applications of the known methods to construct exact solutions of NLPDEs. Mathematical modelling of physical systems often leads t...

This paper develops algorithms for the pricing of discretely sampled barrier, lookback and hindsight options and discretely exercisable American options. Under the Black-Scholes framework, the pricing of these options can be reduced to evaluation of a series of convolutions of the Gaussian distribution and a known function. We compute these convolutions efficiently using the double-exponential ...

Neural networks have shown tremendous growth in recent years to solve numerous problems. Various types of neural been introduced deal with different However, the main goal any network is transform non-linearly separable input data into more linearly abstract features using a hierarchy layers. These layers are combinations linear and nonlinear functions. The most popular common non-linearity act...