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Sato, O.

doi:10.3769/radioisotopes.30.7_410

Cited by 8 publications

(13 citation statements)

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“…It would be also worth clarifying the relation between the Wronskian-type solutions in the present paper and the non-Wronskian-type solutions in [10,11]. Wronskian-type solutions play crucial roles in the Sato theory of the KP equations which reveals hidden infinite symmetry on infinite-dimensional Grassmannians from the viewpoint of Plücker relations of the Wronskians [41]. Hence the present formulation might be a hint for a higher-dimensional extension of Sato theory.…”

Section: Conclusion and Discussionmentioning

confidence: 71%

Soliton solutions of noncommutative anti-self-dual Yang–Mills equations

Gilson¹,

Hamanaka²,

Huang³

et al. 2020

J. Phys. A: Math. Theor.

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We present exact soliton solutions of anti-self-dual Yang-Mills equations for G = GL(N ) on noncommutative Euclidean spaces in four-dimension by using the Darboux transformations. Generated solutions are represented by quasideterminants of Wronski matrices in compact forms. We give special one-soliton solutions for G = GL(2) whose energy density can be real-valued. We find that the soliton solutions are the same as the commutative ones and can be interpreted as one-domain walls in four-dimension. Scattering processes of the multi-soliton solutions are also discussed.

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Section: Conclusion and Discussionmentioning

confidence: 71%

Soliton solutions of noncommutative anti-self-dual Yang–Mills equations

Gilson¹,

Hamanaka²,

Huang³

et al. 2020

J. Phys. A: Math. Theor.

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“…The standard approach to the analysis of the string equation is based on the Sato theory of KP -hierarchy [6] (The most appropriate for our aims exposition of this theory is given in [7].) Our approach is based on more geometrical part of the same theory connected with infinite-dimensional Grassmannian Gr.…”

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confidence: 99%

On Solutions to the String Equation

Schwarz

1991

Mod. Phys. Lett. A

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The set of solutions to the string equation [P, Q] = 1 where P and Q are differential operators is described.It is shown that there exists oneto-one correspondence between this set and the set of pairs of commuting differential operators.This fact permits us to describe the set of solutions to the string equation in terms of moduli spa-ces of algebraic curves,however the direct description is much simpler. Some results are obtained for the superanalog to the string equation where P and Q are considered as super differential operators. It is proved that this equation is invariant with respect to Manin-Radul, Mulase-Rabin and Kac-van de Leur KP-hierarchies. Let us consider ordinary differential operatorswhere ∂ = ∂ ∂x , a m (x) and b n (x) are formal power series with complex coefficients. The approach to non-critical string theory based on the consideration of matrix models [1]-[3] led to the problem of description of all pairs P, Q satisfying [P, Q] = 1 (see [4]). The equation [P, Q] = 1 where P and Q are differential operators is known therefore as string equation. We will assume that the orders p and h of differential operators P and Q have no common divisors and that Q is monic (i.e. the leading coefficient b h (x) is equal to 1). The set of all pairs (P, Q) obeying these conditions will be denoted by A p,h .Every monic operator Q = h r=1 b r (x)∂ r can be normalized by means of transformation Q →Q = e γ Qe −γ (i.e. one can make b h−1 = 0).If (P, Q) ∈ A p,h then (P ,Q) whereP = e γ P e −γ ,Q = e γ Qe −γ belongs to A p,h too. Therefore the study of the space A p,h can be reduced to the study of the space Q p,h consisting of pairs (P, Q) ∈ A p,h where Q is normalized.One of our aims is to describe the set Q p,h . We give the following description. Let us denote by M p,h the space of polynomial h × h matrices P = (P ij (u)) satisfying p = max 1≤j≤h (j − i + h deg P ij ) for every i. (Here i, j = 1, . . . , h, deg P ij denotes the degree of the polynomial P ij (u)).

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“…In [2] and [9,13], we know that the KP equation possesses solutions expressed in a Wronskian form and a Grammian form. Moreover, the coupled KP equation possesses solutions expressed in a Wronski-type Pfaffian and in a Gramm-type Pfaffian.…”

Section: Discussionmentioning

confidence: 99%

“…In [2] and [9][10][11][12][13], the Wronskian and the Grammian determinant solutions for the KP equation were investigated. In [1], the Wronski-type and the Gramm-type Pfaffian solutions for the coupled KP equation were also considered.…”

Section: Introductionmentioning

confidence: 99%