Higher order rogue waves for the(3 + 1)-dimensional Jimbo–Miwa equation

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Mohammed K. Elboree
Keyword(s):  
Symbolic Computation ◽  
Fourth Order ◽  
Bilinear Form ◽  
Rogue Waves ◽  
Higher Order ◽  
Second Order ◽  
Third Order ◽  
First Order ◽  
Hirota Bilinear Form ◽  
Hirota Bilinear

Abstract Based on the Hirota bilinear form for the (3 + 1)-dimensional Jimbo–Miwa equation, we constructed the first-order, second-order, third-order and fourth-order rogue waves for this equation using the symbolic computation approach. Also some properties of the higher-order rogue waves and their interaction are explained by some figures via some special choices of the parameters.

Abundant rogue wave solutions for the (2 + 1)-dimensional generalized Korteweg–de Vries equation

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Huanhuan Lu ◽  
Yufeng Zhang
Keyword(s):  
Vries Equation ◽  
Rogue Wave ◽  
Rogue Waves ◽  
Third Order ◽  
First Order ◽  
Wave Solutions ◽  
De Vries ◽  
Rogue Wave Solutions ◽  
Bilinear Formulation ◽  
Hirota Bilinear

AbstractIn this paper, we analyse two types of rogue wave solutions generated from two improved ansatzs, to the (2 + 1)-dimensional generalized Korteweg–de Vries equation. With symbolic computation, the first-order rogue waves, second-order rogue waves, third-order rogue waves are generated directly from the first ansatz. Based on the Hirota bilinear formulation, another type of one-rogue waves and two-rogue waves can be obtained from the second ansatz. In addition, the dynamic behaviours of obtained rogue wave solutions are illustrated graphically.

Lump-type solutions for the (4+1)-dimensional Fokas equation via symbolic computations

2017 ◽  
Vol 31 (25) ◽  
pp. 1750224 ◽  
Author(s):  
Li Cheng ◽  
Yi Zhang
Keyword(s):  
Symbolic Computation ◽  
Bilinear Form ◽  
Symbolic Computations ◽  
Hirota Bilinear Form ◽  
Free Parameters ◽  
Almost All ◽  
Hirota Bilinear

Based on the Hirota bilinear form, two classes of lump-type solutions of the (4[Formula: see text]+[Formula: see text]1)-dimensional nonlinear Fokas equation, rationally localized in almost all directions in the space are obtained through a direct symbolic computation with Maple. The resulting lump-type solutions contain free parameters. To guarantee the analyticity and rational localization of the solutions, the involved parameters need to satisfy certain constraints. A few particular lump-type solutions with special choices of the involved parameters are given.

scholarly journals Solitons, Rogues and Interaction Behaviors of Third-Order Nonlinear Schrodinger Equation

2022 ◽  
Author(s):  
Ren Bo ◽  
Shi Kai-Zhong ◽  
Shou-Feng Shen ◽  
Wang Guo-Fang ◽  
Peng Jun-Da ◽  
...  
Keyword(s):  
Bilinear Form ◽  
Rogue Wave ◽  
Soliton Solutions ◽  
Ultrashort Pulses ◽  
Rogue Waves ◽  
Third Order ◽  
First Order ◽  
The Third ◽  
Femtosecond Regime ◽  
Wave Limit

Abstract In this paper, we investigate the third-order nonlinear Schr\"{o}dinger equation which is used to describe the propagation of ultrashort pulses in the subpicosecond or femtosecond regime. Based on the independent transformation, the bilinear form of the third-order NLSE is constructed. The multiple soliton solutions are constructed by solving the bilinear form. The multi-order rogue waves and interaction between one-soliton and first-order rogue wave are obtained by the long wave limit in multi-solitons. The dynamics of the first-order rogue wave, second-order rogue wave and interaction between one-soliton and first-order rogue wave are presented by selecting the appropriate parameters. In particular parameters, the positions and the maximum of amplitude of rogue wave can be confirmed by the detail calculations.PACS numbers: 02.30.Ik, 05.45.Yv.

Development of the female flower and gynecandrous partial inflorescence of Myrica californica

1979 ◽  
Vol 57 (2) ◽  
pp. 141-151 ◽  
Author(s):  
Alastair D. Macdonald
Keyword(s):  
Fourth Order ◽  
Female Flower ◽  
Second Order ◽  
Third Order ◽  
First Order ◽  
Unique Structure ◽  
Inflorescence Axis ◽  
Definition Of ◽  
Fifth Order ◽  
Partial Inflorescence

Organogenesis of the female flower and gynecandrous partial inflorescence is described. Approximately 25 first-order inflorescence bracts are formed in an acropetal sequence. A second-order inflorescence axis, the partial inflorescence, develops in the axil of each bract. Third-, fourth-, and fifth-order axes arise in the axils of second-, third-, and fourth-order bracts. A gynoecium terminates a second-order axis and sometimes a distal third-order axis. A gynoecium consists of two stigmas and one basal, unitegmic, orthotropous ovule. The wall enclosing the ovule, the circumlocular wall, is comprised distally of gynoecial tissue and proximally of tissue of the inflorescence axis and its appendages. The latter portion of the wall is formed by zonal growth. Androecial members, formed proximal to the gynoecium on the partial inflorescence, are carried onto the circumlocular wall by zonal growth. A stamen may develop from the last-formed primordium before gynoecial inception or from a potentially stigmatic primordium. The papillae of the flower and fruit arise as emergences and from potentially bracteate, axial, and staminate primorida during the development of the circumlocular wall. The term circumlocular wall is used in a neutral sense to describe this unique structure. Since the gynoecium is composed of gynoecial appendages and inflorescence axis and appendages, a functional definition of gynoecium must be expanded to include any tissue, including an inflorescence, that surrounds the ovule(s) and forms the fruit(s).

scholarly journals Lump and Lump-Type Solutions of the Generalized (3+1)-Dimensional Variable-Coefficient B-Type Kadomtsev-Petviashvili Equation

2019 ◽  
Vol 2019 ◽  
pp. 1-5 ◽  
Author(s):  
Yanni Zhang ◽  
Jing Pang
Keyword(s):  
Symbolic Computation ◽  
Bilinear Form ◽  
Variable Coefficient ◽  
Three Dimensional ◽  
Petviashvili Equation ◽  
Hirota Bilinear Form ◽  
Contour Plots ◽  
Dimensional Variable ◽  
Hirota Bilinear

Based on the Hirota bilinear form of the generalized (3+1)-dimensional variable-coefficient B-type Kadomtsev-Petviashvili equation, the lump and lump-type solutions are generated through symbolic computation, whose analyticity can be easily achieved by taking special choices of the involved parameters. The property of solutions is investigated and exhibited vividly by three-dimensional plots and contour plots.

scholarly journals 02. Social Pathologies, Reflexive Pathologies, and the Idea of Higher-Order Disorders

2015 ◽  
Author(s):  
Arto Laitinen
Keyword(s):  
Social Reality ◽  
Social Criticism ◽  
Higher Order ◽  
Second Order ◽  
Third Order ◽  
First Order ◽  
The Third ◽  
The Social ◽  
Order Reflection

This paper critically examines Christopher Zurn’s suggestion mentioned above that various social pathologies (pathologies of ideological recognition, maldistribution, invisibilization, rationality distortions, reification and institutionally forced self-realization) share the structure of being ‘second-order disorders’: that is, that they each entail ‘constitutive disconnects between first-order contents and secondorder reflexive comprehension of those contents, where those disconnects are pervasive and socially caused’ (Zurn, 2011, 345-346). The paper argues that the cases even as discussed by Zurn do not actually match that characterization, but that it would be premature to conclude that they are not thereby social pathologies, or that they do not have a structure in common. It is just that the structure is more complex than originally described, covering pervasive socially caused evils (i) in the social reality, (ii) in the first order experiences and understandings, (iii) in the second order reflection as discussed by Zurn, and also (iv) in the ‘third order’ phenomenon concerning the pre-emptive silencing or nullification of social criticism even before it takes place 

Second- and higher-order logics

2018 ◽  
Author(s):  
Shaughan Lavine
Keyword(s):  
Predicate Logic ◽  
Higher Order ◽  
Second Order ◽  
Order Logic ◽  
First Order Logic ◽  
Third Order ◽  
First Order ◽  
Proof Systems ◽  
The Universe ◽  
Second Order Logic

In first-order predicate logic there are symbols for fixed individuals, relations and functions on a given universe of individuals and there are variables ranging over the individuals, with associated quantifiers. Second-order logic adds variables ranging over relations and functions on the universe of individuals, and associated quantifiers, which are called second-order variables and quantifiers. Sometimes one also adds symbols for fixed higher-order relations and functions among and on the relations, functions and individuals of the original universe. One can add third-order variables ranging over relations and functions among and on the relations, functions and individuals on the universe, with associated quantifiers, and so on, to yield logics of even higher order. It is usual to use proof systems for higher-order logics (that is, logics beyond first-order) that include analogues of the first-order quantifier rules for all quantifiers. An extensional n-ary relation variable in effect ranges over arbitrary sets of n-tuples of members of the universe. (Functions are omitted here for simplicity: remarks about them parallel those for relations.) If the set of sets of n-tuples of members of a universe is fully determined once the universe itself is given, then the truth-values of sentences involving second-order quantifiers are determined in a structure like the ones used for first-order logic. However, if the notion of the set of all sets of n-tuples of members of a universe is specified in terms of some theory about sets or relations, then the universe of a structure must be supplemented by specifications of the domains of the various higher-order variables. No matter what theory one adopts, there are infinitely many choices for such domains compatible with the theory over any infinite universe. This casts doubt on the apparent clarity of the notion of ‘all n-ary relations on a domain’: since the notion cannot be defined categorically in terms of the domain using any theory whatsoever, how could it be well-determined?

Painlevé Integrability and N-Soliton Solution for the Whitham- Broer-Kaup Shallow Water Model Using Symbolic Computation

2008 ◽  
Vol 63 (5-6) ◽  
pp. 253-260 ◽  
Author(s):  
Cheng Zhang ◽  
Bo Tian ◽  
Xiang-Hua Meng ◽  
Xing Lü ◽  
Ke-Jie Cai ◽  
...  
Keyword(s):  
Shallow Water ◽  
Symbolic Computation ◽  
Soliton Solution ◽  
Bilinear Form ◽  
Water Model ◽  
Shallow Water Model ◽  
Graphic Analysis ◽  
Hirota Bilinear Form ◽  
Hirota Bilinear ◽  
Singular Manifolds

With the help of symbolic computation, the Whitham-Broer-Kaup shallow water model is analyzed for its integrability through the Painlev´e analysis. Then, by truncating the Painlevé expansion at the constant level term with two singular manifolds, the Hirota bilinear form is obtained and the corresponding N-soliton solution with graphic analysis is also given. Furthermore, a bilinear auto-Bäcklund transformation is constructed for the Whitham-Broer-Kaup model, from which a one-soliton solution is presented.

scholarly journals Generalised Algebraic Continued Fractions related to Definite Integrals

1958 ◽  
Vol 9 (4) ◽  
pp. 170-182
Author(s):  
L. R. Shenton
Keyword(s):  
Recurrence Relation ◽  
Fourth Order ◽  
Continued Fractions ◽  
Recurrence Formula ◽  
Second Order ◽  
Third Order ◽  
First Order ◽  
Definite Integrals ◽  
Image Position ◽  
Remarkable Relation

The present paper is a continuation of the work initiated in [l]-[5]. In [5] I gave an expansion of the formfor the second order C.F. associated withwhere U8, V8, W8 satisfy a fourth-order recurrence relation, there being a similar expansion for third order C.F.'s. I shall now give simple expressions for U8, V8, W8 (or related forms) in terms of χ2s(Z1), χ2s (Z2), ω2s(Z1), ω2s(Z2), whereand show that there is a remarkable relation between the recurrence formula for the first order C.F. and that satisfied by U3, V3, W3. The generalised form of these results will be stated and proved.

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