Symmetries of Kadomtsev-Petviashvili equation, isomonodromic deformations, and ?nonlinear? generalizations of the special functions of wave catastrophes

A comprehensive study of the negative-order Kadomtsev–Petviashvili (nKP) partial differential equation by Lie group method has been presented. Initially the infinitesimal generators and symmetry reduction, which were obtained by applying the Lie group method on the negative-order Kadomtsev–Petviashvili equation, have been used for constructing the reduced equations. In particular, the traveling wave solutions for the negative-order KP equation have been derived from the reduced equations as an invariant solution. Finally, the extended improved (G′/G) method and the extended tanh method are described and applied in constructing new explicit expressions for the traveling wave solutions. Many new and more general exact solutions are obtained.

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Certain integral involving the product of Srivastava polynomials and special functions

Afrika Matematika ◽

10.1007/s13370-021-00885-7 ◽

2021 ◽

Author(s):

Dinesh Kumar ◽

Frédéric Ayant ◽

Amit Prakash

Keyword(s):

Special Functions

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Two-dimensional leaps in near-critical flow over isolated orography

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2005.1530 ◽

2005 ◽

Vol 461 (2064) ◽

pp. 3747-3763 ◽

Cited By ~ 3

Author(s):

E.R Johnson ◽

G.G Vilenski

Keyword(s):

Unsteady Flow ◽

Equations Of Motion ◽

Fold Increase ◽

Flow Speed ◽

Two Dimensional ◽

Subcritical Flow ◽

One Dimensional ◽

Petviashvili Equation ◽

Lee Wave ◽

Supercritical Flows

This paper describes steady two-dimensional disturbances forced on the interface of a two-layer fluid by flow over an isolated obstacle. The oncoming flow speed is close to the linear longwave speed and the layer densities, layer depths and obstacle height are chosen so that the equations of motion reduce to the forced two-dimensional Korteweg–de Vries equation with cubic nonlinearity, i.e. the forced extended Kadomtsev–Petviashvili equation. The distinctive feature noted here is the appearance in the far lee-wave wake behind obstacles in subcritical flow of a ‘table-top’ wave extending almost one-dimensionally for many obstacles widths across the flow. Numerical integrations show that the most important parameter determining whether this wave appears is the departure from criticality, with the wave appearing in slightly subcritical flows but being destroyed by two-dimensional effects behind even quite long ridges in even moderately subcritical flow. The wave appears after the flow has passed through a transition from subcritical to supercritical over the obstacle and its leading and trailing edges resemble dissipationless leaps standing in supercritical flow. Two-dimensional steady supercritical flows are related to one-dimensional unsteady flows with time in the unsteady flow associated with a slow cross-stream variable in the two-dimensional flows. Thus the wide cross-stream extent of the table-top wave appears to derive from the combination of its occurrence in a supercritical region embedded in the subcritical flow and the propagation without change of form of table-top waves in one-dimensional unsteady flow. The table-top wave here is associated with a resonant steepening of the transition above the obstacle and a consequent twelve-fold increase in drag. Remarkably, the table-top wave is generated equally strongly and extends laterally equally as far behind an axisymmetric obstacle as behind a ridge and so leads to subcritical flows differing significantly from linear predictions.

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Multiple solutions and ground state solutions for a class of generalized Kadomtsev-Petviashvili equation

Open Mathematics ◽

10.1515/math-2021-0014 ◽

2021 ◽

Vol 19 (1) ◽

pp. 297-305

Author(s):

Yuting Zhu ◽

Chunfang Chen ◽

Jianhua Chen ◽

Chenggui Yuan

Keyword(s):

Bounded Domain ◽

Ground State ◽

Multiple Solutions ◽

Nontrivial Solutions ◽

Ground State Solutions ◽

Petviashvili Equation

Abstract In this paper, we study the following generalized Kadomtsev-Petviashvili equation u t + u x x x + ( h ( u ) ) x = D x − 1 Δ y u , {u}_{t}+{u}_{xxx}+{\left(h\left(u))}_{x}={D}_{x}^{-1}{\Delta }_{y}u, where ( t , x , y ) ∈ R + × R × R N − 1 \left(t,x,y)\in {{\mathbb{R}}}^{+}\times {\mathbb{R}}\times {{\mathbb{R}}}^{N-1} , N ≥ 2 N\ge 2 , D x − 1 f ( x , y ) = ∫ − ∞ x f ( s , y ) d s {D}_{x}^{-1}f\left(x,y)={\int }_{-\infty }^{x}f\left(s,y){\rm{d}}s , f t = ∂ f ∂ t {f}_{t}=\frac{\partial f}{\partial t} , f x = ∂ f ∂ x {f}_{x}=\frac{\partial f}{\partial x} and Δ y = ∑ i = 1 N − 1 ∂ 2 ∂ y i 2 {\Delta }_{y}={\sum }_{i=1}^{N-1}\frac{{\partial }^{2}}{{\partial }_{{y}_{i}}^{2}} . We get the existence of infinitely many nontrivial solutions under certain assumptions in bounded domain without Ambrosetti-Rabinowitz condition. Moreover, by using the method developed by Jeanjean [13], we establish the existence of ground state solutions in R N {{\mathbb{R}}}^{N} .

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Circular quiver gauge theories, isomonodromic deformations and $$W_N$$ fermions on the torus

Letters in Mathematical Physics ◽

10.1007/s11005-020-01343-4 ◽

2021 ◽

Vol 111 (3) ◽

Author(s):

Giulio Bonelli ◽

Fabrizio Del Monte ◽

Pavlo Gavrylenko ◽

Alessandro Tanzini

Keyword(s):

Integrable System ◽

Gauge Theories ◽

The Self ◽

Free Fermion ◽

Two Dimensional ◽

Isomonodromic Deformation ◽

Specific Time ◽

Dimensional Torus ◽

Isomonodromic Deformations ◽

Deformation Problem

AbstractWe study the relation between class $$\mathcal {S}$$ S theories on punctured tori and isomonodromic deformations of flat SL(N) connections on the two-dimensional torus with punctures. Turning on the self-dual $$\Omega $$ Ω -background corresponds to a deautonomization of the Seiberg–Witten integrable system which implies a specific time dependence in its Hamiltonians. We show that the corresponding $$\tau $$ τ -function is proportional to the dual gauge theory partition function, the proportionality factor being a nontrivial function of the solution of the deautonomized Seiberg–Witten integrable system. This is obtained by mapping the isomonodromic deformation problem to $$W_N$$ W N free fermion correlators on the torus.

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New fractional approaches for n-polynomial P-convexity with applications in special function theory

Advances in Difference Equations ◽

10.1186/s13662-020-03000-5 ◽

2020 ◽

Vol 2020 (1) ◽

Cited By ~ 1

Author(s):

Shu-Bo Chen ◽

Saima Rashid ◽

Muhammad Aslam Noor ◽

Zakia Hammouch ◽

Yu-Ming Chu

Keyword(s):

Fractional Calculus ◽

Convex Functions ◽

Special Functions ◽

Real Life ◽

Fractional Integrals ◽

Digamma Function ◽

Novel Approach ◽

Inequality Theory ◽

New Strategy ◽

Significant Mechanism

Abstract Inequality theory provides a significant mechanism for managing symmetrical aspects in real-life circumstances. The renowned distinguishing feature of integral inequalities and fractional calculus has a solid possibility to regulate continuous issues with high proficiency. This manuscript contributes to a captivating association of fractional calculus, special functions and convex functions. The authors develop a novel approach for investigating a new class of convex functions which is known as an n-polynomial $\mathcal{P}$ P -convex function. Meanwhile, considering two identities via generalized fractional integrals, provide several generalizations of the Hermite–Hadamard and Ostrowski type inequalities by employing the better approaches of Hölder and power-mean inequalities. By this new strategy, using the concept of n-polynomial $\mathcal{P}$ P -convexity we can evaluate several other classes of n-polynomial harmonically convex, n-polynomial convex, classical harmonically convex and classical convex functions as particular cases. In order to investigate the efficiency and supremacy of the suggested scheme regarding the fractional calculus, special functions and n-polynomial $\mathcal{P}$ P -convexity, we present two applications for the modified Bessel function and $\mathfrak{q}$ q -digamma function. Finally, these outcomes can evaluate the possible symmetric roles of the criterion that express the real phenomena of the problem.

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