Nonlinear equations in soliton physics and operator ideals

Carl, Bernd; Schiebold, Cornelia

doi:10.1088/0951-7715/12/2/012

Cited by 52 publications

(61 citation statements)

References 19 publications

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“…For example, if S is chosen to have complex eigenvalues, then, for any real ω in Γ k,N , τ S ω is a real KP tau-function of a complexiton type. 6 It has already been noted in other contexts that n-soliton solutions "live" in finite dimensional Grassmannians [16,19].…”

Section: 2mentioning

confidence: 94%

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On KP generators and the geometry of the HBDE

Gekhtman

Kasman

2006

Journal of Geometry and Physics

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Abstract. Sato theory provides a correspondence between solutions to the KP hierarchy and points in an infinite dimensional Grassmannian. In this correspondence, flows generated infinitesimally by powers of the "shift" operator give time dependence to the first coordinate of an arbitrarily selected point, making it a tau-function. These tau-functions satisfy a number of integrable equations, including the Hirota Bilinear Difference Equation (HBDE). Here, we rederive the HBDE as a statement about linear maps between Grassmannians. In addition to illustrating the fundamental nature of this equation in the standard theory, we make use of this geometric interpretation of the HBDE to answer the question of what other infinitesimal generators could be used for similarly creating tau-functions. The answer to this question involves a "rank one condition", tying this investigation to the existing results on integrable systems involving such conditions and providing an interpretation for their significance in terms of the relationship between the HBDE and the geometry of Grassmannians.

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Section: 2mentioning

confidence: 94%

“…The matrices X, Y and Z can be selected so as to make τ S ω (t) the tau-function of an n-soliton solution to the KP hierarchy 6 by choosing 4n complex parameters µ i , λ i , α i and γ i (1 ≤ i ≤ n, such that µ i = λ j ) and letting…”

Section: 2mentioning

confidence: 99%

On KP generators and the geometry of the HBDE

Gekhtman

Kasman

2006

Journal of Geometry and Physics

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“…Actually, the rank one condition appeared in many literatures (e.g. [1,[8][9][10][23][24][25]28,45,47,51]). Note that any N-th order rank one matrix is similar to the product rs T with suitable N-th order vectors r and s. Recalling our master function S (i, j) = s T K j (I + M) −1 K i r, the rank one condition naturally guarantees (or comes from the fact) S (i, j) defines a scaler function.…”

Section: Deformation Of the Sylvester Equation And Rank One Conditionmentioning

confidence: 99%

“…This method needs lengthy verification of solutions. For the review of this method one can see [9,10]. This method relies on the Sylvester equation (1.1) with C of rank one so as to get needed trace property.…”

Section: Introductionmentioning

confidence: 99%

The Sylvester equation and integrable equations: I. The Korteweg-de Vries system and sine-Gordon equation

Xu¹,

Zhang²,

Zhao³

2021

JNMP

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The paper is to reveal the direct links between the well known Sylvester equation in matrix theory and some integrable systems. Using the Sylvester equation KM + MK = r s T we introduce a scalar functionwhich is defined as same as in discrete case. S (i, j) satisfy some recurrence relations which can be viewed as discrete equations and play indispensable roles in deriving continuous integrable equations. By imposing dispersion relations on r and s, we find the Korteweg-de Vries equation, modified Korteweg-de Vries equation, Schwarzian Korteweg-de Vries equation and sine-Gordon equation can be expressed by some discrete equations of S (i, j) defined on certain points. Some special matrices are used to solve the Sylvester equation and prove symmetry property S (i, j) = S ( j,i) . The solution M provides τ function by τ = |I + M|. We hope our results can not only unify the Cauchy matrix approach in both continuous and discrete cases, but also bring more links for integrable systems and variety of areas where the Sylvester equation appears frequently.

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“…In [3] the result was obtained under the additional assumption [A, B] = 0 (see also [11]), which is very restrictive in applications. One of our motivations were similar formulas as of Sakhnovich [10], which were only verified by means of computer algebra.…”

Section: Then On = {(X Y T) ∈ ‫ޒ‬ 3 | (I + Ml) Is Invertible} a Smentioning

confidence: 99%

Explicit Solution Formulas for the Matrix-Kp

Schiebold¹

2009

Glasgow Math. J.

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Abstract. We study a non-commutative version of the Kadomtsev-Petviashvili equations and construct a family of solutions generalizing naturally the soliton to the non-commutative setting. From this we derive explicit solution formulas as well for the scalar as for the matrix-Kadomtsev-Petviashvili equation which still depend on operator parameters.2000 Mathematics Subject Classification. 39B42, 47J25, 37K10, 35C05.

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Nonlinear equations in soliton physics and operator ideals

Cited by 52 publications

References 19 publications

On KP generators and the geometry of the HBDE

On KP generators and the geometry of the HBDE

The Sylvester equation and integrable equations: I. The Korteweg-de Vries system and sine-Gordon equation

Explicit Solution Formulas for the Matrix-Kp

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