An Elementary Introduction to Sato Theory

Ohta, Yuichi; Satsuma, Junkichi; Takahashi, Daisuke; Tokihiro, Tetsuji

doi:10.1143/ptps.94.210

Cited by 237 publications

(174 citation statements)

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“…This material also can be related to pseudodifferential operators (PSDO) directly with a Lax operator and gauge operator IV, where L = WOW -1 and c~,,WW -t = -L~ (Sato equation; cf. [8,56,60,76]). There is no need here to involve a spectral parameter A (although at times this is expedient and convenient).…”

Section: Introductionmentioning

confidence: 99%

Remarks on dispersionless KP, KdV, and 2D gravity

Carroll

1994

J Nonlinear Sci

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Summary. We use the SDiff(2) framework of Takasaki and Takebe and the (L, M) program (L is the Lax operator and Mq~ = Ox) to show that ~02 = semiclassical limit of M is ~+L~2T'A n-l, where (A, -~) are action angle variables in the Gibbons-Kodama theory of Hamilton-Jacobi type for dispersionless KP. We also show ~ is the semiclassical limit of WxW -1 (W is the gauge operator), where G = WxW -1 is a quantity studied by the author in an earlier paper in connection with symmetries. We give then a semiclassical version of the Jevicki-Yoneya action principle for 2D gravity, where again ~ arises in calculations, and this yields directly the Landau-Ginsburg equation that corresponds to the semiclassical limit of an integrated string equation. For KdV we also show how inverse scattering data are connected to Hamiltonians for dispersionless KdV. We also discuss Hirota bilinear formulas relative to the dispersionless hierarchies and establish various limiting formulas.

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Section: Introductionmentioning

confidence: 99%

Remarks on dispersionless KP, KdV, and 2D gravity

Carroll

1994

J Nonlinear Sci

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“…Based on the framework of the Sato theory, we first give a simple description of the mKP hierarchy [29][30][31]. Let us consider the following pseudo-differential operator (PDO)…”

Section: The Darboux Transformation For Mkp Equationmentioning

confidence: 99%

Some types of solutions and generalized binary Darboux transformation for the mKP equation with self-consistent sources

Tian

Wang

Zhang

2010

Journal of Mathematical Analysis and Applications

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In this paper, an mKP equation with self-consistent sources (mKPESCSs) is structured in the framework of the constrained mKP equation. Based on the conjugate Lax pairs, we construct the generalized binary Darboux transformation and the N-times repeated Darboux transformation with arbitrary functions at time t for the mKPESCSs which offers a non-auto-Bäcklund transformation between two mKPESCSs with different degrees of sources. With the help of these transformations, some new solutions for the mKPESCSs such as soliton solutions, rational solutions, breather type solutions and exponential solutions are found by taking the special initial solution for auxiliary linear problems and the special functions of t-time.

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“…Discussion of all this goes beyond the scope of this article. For connections to Grassmannians etc., we refer to [45,63,95,100,101,133] (el. also Remark 5.6).…”

Section: Complementsmentioning

confidence: 99%

On the ubiquitous Gelfand-Levitan-Mar?enko (GLM) equation

Carroll

1990

Acta Appl Math

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We collect a number of derivations and interpretations of Gelfand-Levitan-Marrenko (GLM) equations in various contexts. Relations to Riemann-Hilbert (RH) problems, tau functions and determinants, direct linearization techniques, etc. are discussed and some new results are included. We emphasize the role of spectral pairings of generalized eigenfunctions and exhibit various ways in which inverse scattering techniques arise. The natural extension and generalization of GLM techniques in the format of RH methods and coadjoint orbits in a Lie theoretic context is developed and the hierarchy point of view is explored in terms of spectral ideas. Thus, the roles and some limitations of the GLM and RH methods are exposed to some extent and some bridges are sketched to modern techniques in soliton mathematics involving hierarchies, tau functions, Lie theory, etc.

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An Elementary Introduction to Sato Theory

Cited by 237 publications

References 0 publications

Remarks on dispersionless KP, KdV, and 2D gravity

Remarks on dispersionless KP, KdV, and 2D gravity

Some types of solutions and generalized binary Darboux transformation for the mKP equation with self-consistent sources

On the ubiquitous Gelfand-Levitan-Mar?enko (GLM) equation

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