Rational solutions to two Sawada–Kotera-like equations

2019 ◽  
Vol 33 (09) ◽  
pp. 1950108
Author(s):  
Ya-Hong Du ◽  
Yin-Shan Yun ◽  
Wen-Xiu Ma
Keyword(s):  
Prime Numbers ◽  
Rational Solutions ◽  
Polynomial Solutions ◽  
Bilinear Operators ◽  
Bilinear Equations

Two Sawada–Kotera-like equations are introduced by the generalized bilinear operators [Formula: see text] associated with two prime numbers [Formula: see text] and [Formula: see text], respectively. Rational solutions of the two presented Sawada–Kotera-like equations are generated by searching polynomial solutions of the corresponding two generalized bilinear equations.

scholarly journals ON PROPERTIES OF FINITE-ORDER MEROMORPHIC SOLUTIONS OF A CERTAIN DIFFERENCE PAINLEVÉ I EQUATION

2011 ◽  
Vol 85 (3) ◽  
pp. 463-475 ◽  
Author(s):  
MEI-RU CHEN ◽  
ZONG-XUAN CHEN
Keyword(s):  
Finite Order ◽  
Meromorphic Solutions ◽  
Rational Solutions ◽  
Polynomial Solutions ◽  
The Difference ◽  
Image Position

AbstractIn this paper, we investigate properties of finite-order transcendental meromorphic solutions, rational solutions and polynomial solutions of the difference Painlevé I equation where a, b and c are constants, ∣a∣+∣b∣≠0.

A Study on Rational Solutions to a KP-like Equation

2015 ◽  
Vol 70 (4) ◽  
pp. 263-268 ◽  
Author(s):  
Yufeng Zhang ◽  
Wen-Xiu Ma
Keyword(s):  
Differential Equation ◽  
Prime Number ◽  
Nonlinear Differential Equation ◽  
Symbolic Computation ◽  
Bilinear Form ◽  
Differential Operators ◽  
Rational Solutions ◽  
Polynomial Solutions ◽  
Bilinear Equation

AbstractA KP-like nonlinear differential equation is introduced through a generalised bilinear equation which possesses the same bilinear form as the standard KP bilinear equation. By symbolic computation, nine classes of rational solutions to the resulting KP-like equation are generated from a search for polynomial solutions to the corresponding generalised bilinear equation. Three generalised bilinear differential operators adopted are associated with the prime number p=3.

Restrictions on meromorphic solutions of Fermat type equations

2020 ◽  
Vol 63 (3) ◽  
pp. 654-665
Author(s):  
Gary G. Gundersen ◽  
Katsuya Ishizaki ◽  
Naofumi Kimura
Keyword(s):  
Meromorphic Functions ◽  
Entire Functions ◽  
Existence Theorems ◽  
Rational Functions ◽  
Meromorphic Solutions ◽  
Entire Solutions ◽  
Rational Solutions ◽  
Polynomial Solutions ◽  
Nevanlinna Functions ◽  
Positive Integers

AbstractThe Fermat type functional equations $(*)\, f_1^n+f_2^n+\cdots +f_k^n=1$, where n and k are positive integers, are considered in the complex plane. Our focus is on equations of the form (*) where it is not known whether there exist non-constant solutions in one or more of the following four classes of functions: meromorphic functions, rational functions, entire functions, polynomials. For such equations, we obtain estimates on Nevanlinna functions that transcendental solutions of (*) would have to satisfy, as well as analogous estimates for non-constant rational solutions. As an application, it is shown that transcendental entire solutions of (*) when n = k(k − 1) with k ≥ 3, would have to satisfy a certain differential equation, which is a generalization of the known result when k = 3. Alternative proofs for the known non-existence theorems for entire and polynomial solutions of (*) are given. Moreover, some restrictions on degrees of polynomial solutions are discussed.

A class of nonlinear heat-type equations and their multi-wave solutions by a generalized bilinear technique

2016 ◽  
Vol 30 (06) ◽  
pp. 1650067
Author(s):  
Jun-Rong Liu
Keyword(s):  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
New Technique ◽  
Bilinear Operators ◽  
Wave Solutions ◽  
Dependent Variables ◽  
A New Technique ◽  
Bilinear Equations

Multi-wave resonant solutions of a class of nonlinear heat-type equations are investigated by their corresponding generalized bilinear equations. We develop a new technique for searching for resonant solutions to those generalized bilinear equations, using the idea of weights of dependent variables. The results show that generalized bilinear operators and generalized bilinear equations are powerful and irreplaceable tools for dealing with nonlinear differential equations.

Lump-type solutions to nonlinear differential equations derived from generalized bilinear equations

2016 ◽  
Vol 30 (28n29) ◽  
pp. 1640018 ◽  
Author(s):  
Wen-Xiu Ma ◽  
Yuan Zhou ◽  
Rachael Dougherty
Keyword(s):  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Symbolic Computations ◽  
Polynomial Solutions ◽  
Quartic Polynomial ◽  
Dependent Variables ◽  
Bilinear Equations

Lump-type solutions, rationally localized in many directions in the space, are analyzed for nonlinear differential equations derived from generalized bilinear differential equations. By symbolic computations with Maple, positive quadratic and quartic polynomial solutions to two classes of generalized bilinear differential equations on [Formula: see text] are computed, and thus, lump-type solutions are presented to the corresponding nonlinear differential equations on [Formula: see text], generated from taking a transformation of dependent variables [Formula: see text].

A remark on a theorem of the Goldbach-Waring type

2004 ◽  
Vol 41 (3) ◽  
pp. 309-324
Author(s):  
C. Bauer
Keyword(s):  
Prime Numbers

Let pi, 2 ≤ i ≤ 5 be prime numbers. It is proved that all but ≪ x23027/23040+ε even integers N ≤ x can be written as N = p21 + p32 + p43 + p45.

The First 50 Million Prime Numbers

1977 ◽  
Vol 1 (S2) ◽  
pp. 7-19 ◽  
Author(s):  
Don Zagier
Keyword(s):  
Prime Numbers

Density of summable subsequences of a sequence and its applications

2020 ◽  
Vol 70 (3) ◽  
pp. 657-666
Author(s):  
Bingzhe Hou ◽  
Yue Xin ◽  
Aihua Zhang
Keyword(s):  
Prime Numbers ◽  
Linear Operators ◽  
Upper Density ◽  
Asymptotic Densities

AbstractLet x = $\begin{array}{} \displaystyle \{x_n\}_{n=1}^{\infty} \end{array}$ be a sequence of positive numbers, and 𝓙x be the collection of all subsets A ⊆ ℕ such that $\begin{array}{} \displaystyle \sum_{k\in A} \end{array}$xk < +∞. The aim of this article is to study how large the summable subsequence could be. We define the upper density of summable subsequences of x as the supremum of the upper asymptotic densities over 𝓙x, SUD in brief, and we denote it by D*(x). Similarly, the lower density of summable subsequences of x is defined as the supremum of the lower asymptotic densities over 𝓙x, SLD in brief, and we denote it by D*(x). We study the properties of SUD and SLD, and also give some examples. One of our main results is that the SUD of a non-increasing sequence of positive numbers tending to zero is either 0 or 1. Furthermore, we obtain that for a non-increasing sequence, D*(x) = 1 if and only if $\begin{array}{} \displaystyle \liminf_{k\to\infty}nx_n=0, \end{array}$ which is an analogue of Cauchy condensation test. In particular, we prove that the SUD of the sequence of the reciprocals of all prime numbers is 1 and its SLD is 0. Moreover, we apply the results in this topic to improve some results for distributionally chaotic linear operators.

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