Logarithmic-type and Exponential-type Hypergeometric Functions for Function Fields

This paper is devoted to the study of a (3 + 1)-dimensional generalized nonlinear evolution equation for the shallow-water waves. The breather solutions with different structures are obtained based on the bilinear form with perturbation parameters. Some new lump solitons are found in the process of studying the degradation behavior of breather solutions, and we also study general lump soliton, lumpoff solution and superposition phenomenon between lump soliton and breather solution. Besides, some theorems about the superposition between lump soliton and [Formula: see text]-soliton ([Formula: see text] is a nonnegative integer) are given. Some examples, including lump-[Formula: see text]-exponential type, lump-[Formula: see text]-logarithmic type, higher-order lump-type [Formula: see text]-soliton, are given to illustrate the correctness of the theorems and corollaries described. Finally, some novel nonlinear phenomena, such as emergence of lump soliton, degeneration of breathers, fission and fusion of lumpoff, superposition of lump-[Formula: see text]-solitons, etc., are analyzed and simulated.

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Hypergeometric Functions and Carlitz Differential Equations over Function Fields

Arithmetic and Geometry Around Hypergeometric Functions - Progress in Mathematics ◽

10.1007/978-3-7643-8284-1_6 ◽

2007 ◽

pp. 163-187

Author(s):

Anatoly N. Kochubei

Keyword(s):

Differential Equations ◽

Hypergeometric Functions ◽

Function Fields

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Asymptotic expansions and converging factors IV. Confluent hypergeometric, parabolic cylinder, modified Bessel, and ordinary Bessel functions

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1959.0022 ◽

1959 ◽

Vol 249 (1257) ◽

pp. 270-283 ◽

Cited By ~ 6

Keyword(s):

Exponential Type ◽

Asymptotic Expansions ◽

Hypergeometric Functions ◽

Bessel Functions ◽

Parabolic Cylinder ◽

Modified Bessel Functions ◽

Confluent Hypergeometric Functions ◽

Special Cases ◽

Parabolic Cylinder Functions

A theory of confluent hypergeometric functions is developed, based upon the methods described in the first three papers (I, II and III) of this series for replacing the divergent parts of asymptotic expansions by easily calculable series involving one or other of the four ‘ basic converging factors ’ which were investigated and tabulated in I. This theory is then illustrated by application to the special cases of exponential-type integrals, parabolic cylinder functions, modified Bessel functions, and ordinary Bessel functions.

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ABEL ARITHMETIC FUNCTIONS IN FINITE CHARACTERISTIC

International Journal of Number Theory ◽

10.1142/s1793042112500959 ◽

2012 ◽

Vol 08 (06) ◽

pp. 1557-1568

Author(s):

DAVID ADAM

Keyword(s):

Exponential Type ◽

Positive Characteristic ◽

Arithmetic Function ◽

Function Fields ◽

Arithmetic Functions ◽

Finite Characteristic

We define Abel arithmetic functions in function fields with positive characteristic. We prove that any Abel arithmetic function with exponential type lesser than [Formula: see text] is a polynomial, the bound [Formula: see text] being optimal. This provides an analog of a Bertrandias' result. Some q-analogs are considered too.

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Singular weighted sharp Trudinger-Moser inequalities defined on $ \mathbb{R}^N $ and applications to elliptic nonlinear equations

Discrete and Continuous Dynamical Systems ◽

10.3934/dcds.2021137 ◽

2021 ◽

Vol 0 (0) ◽

pp. 0

Author(s):

Sami Aouaoui ◽

Rahma Jlel

Keyword(s):

Euclidean Space ◽

Nonlinear Equations ◽

Exponential Type ◽

Growth Conditions ◽

Concentration Compactness ◽

Weighted Inequalities ◽

Radial Functions ◽

Compactness Arguments ◽

Logarithmic Type ◽

Conditions At Infinity

<p style='text-indent:20px;'>This work comes to complete some previous ones of ours. Actually, in this paper, we establish some singular weighted inequalities of Trudinger-Moser type for radial functions defined on the whole euclidean space <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^N,\ N \geq 2. $\end{document}</tex-math></inline-formula> The weights considered are of logarithmic type. The singularity plays a capital role to prove the sharpness of the inequalities. These inequalities are later improved using some concentration-compactness arguments. The last part in this work is devoted to the application of the inequalities established to some singular elliptic nonlinear equations involving a new growth conditions at infinity of exponential type.</p>

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