Constrained KP hierarchy as a ratio of differential operators

2007 ◽  
pp. 2-17
Author(s):  
Henrik Aratyn
Keyword(s):  
Differential Operators ◽  
Kp Hierarchy

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.

The Calder´on-Zygmund Theory I: Ellipticity

2017 ◽  
Author(s):  
Brian Street
Keyword(s):  
Classical Theory ◽  
Singular Integral ◽  
Singular Integrals ◽  
Differential Operators ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Single Parameter ◽  
Partial Differential ◽  
Translation Invariant ◽  
Partial Differential Operators

This chapter discusses a case for single-parameter singular integral operators, where ρ‎ is the usual distance on ℝn. There, we obtain the most classical theory of singular integrals, which is useful for studying elliptic partial differential operators. The chapter defines singular integral operators in three equivalent ways. This trichotomy can be seen three times, in increasing generality: Theorems 1.1.23, 1.1.26, and 1.2.10. This trichotomy is developed even when the operators are not translation invariant (many authors discuss such ideas only for translation invariant, or nearly translation invariant operators). It also presents these ideas in a slightly different way than is usual, which helps to motivate later results and definitions.

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
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