Pseudo-differential operators and the KP hierarchy

2010 ◽  
pp. 219-234
Author(s):  
Alex Kasman
Keyword(s):  
Differential Operators ◽  
Kp Hierarchy ◽  
Pseudo Differential Operators

The Extended C-Type of KP Hierarchy: Non-Auto Darboux Transformation and Solutions

2016 ◽  
Vol 71 (10) ◽  
pp. 933-941
Author(s):  
Hongxia Wu ◽  
Chunxia Li ◽  
Yunbo Zeng
Keyword(s):  
Darboux Transformation ◽  
Differential Operators ◽  
Explicit Solutions ◽  
Kp Hierarchy ◽  
Pseudo Differential Operators ◽  
Self Consistent ◽  
Ckp Hierarchy

AbstractThe pseudo-differential operators are used to construct the non-auto Darboux transformation (DT) for extended C-type of KP (CKP) hierarchy and the corresponding generalised Wronskian solutions are derived. In addition, explicit solutions of soliton-type are formulated for the second type of CKP equation with self-consistent sources (CKPESCS).

scholarly journals AMBROSE–SINGER THEOREM ON DIFFEOLOGICAL BUNDLES AND COMPLETE INTEGRABILITY OF THE KP EQUATION

2013 ◽  
Vol 10 (09) ◽  
pp. 1350043 ◽  
Author(s):  
JEAN-PIERRE MAGNOT
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Differential Operators ◽  
Complete Integrability ◽  
Structure Group ◽  
Connection Form ◽  
Infinite Dimensions ◽  
Kp Hierarchy ◽  
Pseudo Differential Operators ◽  
Petviashvili Equation

In this paper, we start from an extension of the notion of holonomy on diffeological bundles, reformulate the notion of regular Lie group or Frölicher Lie groups, state an Ambrose–Singer theorem that enlarges the one stated in [J.-P. Magnot, Structure groups and holonomy in infinite dimensions, Bull. Sci. Math.128 (2004) 513–529], and conclude with a differential geometric treatment of KP hierarchy. The examples of Lie groups that are studied are principally those obtained by enlarging some graded Frölicher (Lie) algebras such as formal q-series of the quantum algebra of pseudo-differential operators. These deformations can be defined for classical pseudo-differential operators but they are used here on formal pseudo-differential operators in order to get a differential geometric framework to deal with the KP hierarchy that is known to be completely integrable with formal power series. Here, we get an integration of the Zakharov–Shabat connection form by means of smooth sections of a (differential geometric) bundle with structure group, some groups of q-deformed operators. The integration obtained by Mulase [Complete integrability of the Kadomtsev–Petviashvili equation Adv. Math.54 (1984) 57–66], and the key tools he developed, are totally recovered on the germs of the smooth maps of our construction. The tool coming from (classical) differential geometry used in this construction is the holonomy group, on which we have an Ambrose–Singer-like theorem: the Lie algebra is spanned by the curvature elements. This result is proved for any connection a diffeological principal bundle with structure group a regular Frölicher Lie group. The case of a (classical) Lie group modeled on a complete locally convex topological vector space is also recovered and the work developed in [J.-P. Magnot, Difféologie du fibré d'Holonomie en dimension infinie, Math. Rep. Canadian Roy. Math. Soc.28(4) (2006); J.-P. Magnot, Structure groups and holonomy in infinite dimensions, Bull. Sci. Math. 128 (2004) 513–529] is completed.

Pseudo-differential operators involving Fractional Fourier cosine (sine) transform

Filomat ◽  
2017 ◽  
Vol 31 (6) ◽  
pp. 1791-1801
Author(s):  
Akhilesh Prasad ◽  
Manoj Singh
Keyword(s):  
Differential Operators ◽  
Fourier Cosine ◽  
Sine Transform ◽  
Pseudo Differential Operators

scholarly journals A Radial Basis Function Finite Difference Scheme for the Benjamin–Ono Equation

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 65
Author(s):  
Benjamin Akers ◽  
Tony Liu ◽  
Jonah Reeger
Keyword(s):  
Radial Basis Function ◽  
Hilbert Transform ◽  
Basis Function ◽  
Differential Operators ◽  
Convergence Rates ◽  
Traveling Wave Solutions ◽  
Algebraic Decay ◽  
Pseudo Differential Operators ◽  
Radial Basis ◽  
The Hilbert Transform

A radial basis function-finite differencing (RBF-FD) scheme was applied to the initial value problem of the Benjamin–Ono equation. The Benjamin–Ono equation has traveling wave solutions with algebraic decay and a nonlocal pseudo-differential operator, the Hilbert transform. When posed on R, the former makes Fourier collocation a poor discretization choice; the latter is challenging for any local method. We develop an RBF-FD approximation of the Hilbert transform, and discuss the challenges of implementing this and other pseudo-differential operators on unstructured grids. Numerical examples, simulation costs, convergence rates, and generalizations of this method are all discussed.

Boundedness of multi-parameter pseudo-differential operators on multi-parameter local Hardy spaces

2020 ◽  
Vol 32 (4) ◽  
pp. 919-936 ◽  
Author(s):  
Jiao Chen ◽  
Wei Ding ◽  
Guozhen Lu
Keyword(s):  
Hardy Spaces ◽  
Differential Operators ◽  
Integral Operators ◽  
Fourier Multipliers ◽  
Regularity Of Solutions ◽  
Local Version ◽  
Fourier Integral Operators ◽  
Pseudo Differential Operators ◽  
Local Hardy Spaces ◽  
The One

AbstractAfter the celebrated work of L. Hörmander on the one-parameter pseudo-differential operators, the applications of pseudo-differential operators have played an important role in partial differential equations, geometric analysis, harmonic analysis, theory of several complex variables and other branches of modern analysis. For instance, they are used to construct parametrices and establish the regularity of solutions to PDEs such as the {\overline{\partial}} problem. The study of Fourier multipliers, pseudo-differential operators and Fourier integral operators has stimulated further such applications. It is well known that the one-parameter pseudo-differential operators are {L^{p}({\mathbb{R}^{n}})} bounded for {1<p<\infty}, but only bounded on local Hardy spaces {h^{p}({\mathbb{R}^{n}})} introduced by Goldberg in [D. Goldberg, A local version of real Hardy spaces, Duke Math. J. 46 1979, 1, 27–42] for {0<p\leq 1}. Though much work has been done on the {L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} boundedness for {1<p<\infty} and Hardy {H^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} boundedness for {0<p\leq 1} for multi-parameter Fourier multipliers and singular integral operators, not much has been done yet for the boundedness of multi-parameter pseudo-differential operators in the range of {0<p\leq 1}. The main purpose of this paper is to establish the boundedness of multi-parameter pseudo-differential operators on multi-parameter local Hardy spaces {h^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} for {0<p\leq 1} recently introduced by Ding, Lu and Zhu in [W. Ding, G. Lu and Y. Zhu, Multi-parameter local Hardy spaces, Nonlinear Anal. 184 2019, 352–380].

scholarly journals On the Weyl transform for rotationally invariant symbols

2019 ◽  
Vol 10 (4) ◽  
pp. 769-791 ◽  
Author(s):  
Norbert Ortner ◽  
Peter Wagner
Keyword(s):  
Asymptotic Behavior ◽  
Differential Operators ◽  
Symmetric Distributions ◽  
Yield Criteria ◽  
Weyl Transform ◽  
Pseudo Differential Operators ◽  
Weyl Transforms ◽  
Rotationally Invariant

Abstract Several formulas for the eigenvalues $$\lambda _j$$ λ j of the Weyl transforms $$W_\sigma $$ W σ of symbols $$\sigma $$ σ given by radially symmetric distributions are derived. These yield criteria for the boundedness and the compactness, respectively, of the pseudo-differential operators $$W_\sigma .$$ W σ . We investigate some examples by analyzing the asymptotic behavior of $$\lambda _j$$ λ j for $$j\rightarrow \infty $$ j → ∞ .

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