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The algebraic structure of the non-commutative nonlinear Schrodinger and modified Korteweg-de Vries hierarchy
Exactly Solvable and Integrable Systems · Submitted 13 Mar 2023
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We prove that each member of the non-commutative nonlinear Schrodinger and modified Korteweg--de Vries hierarchy is a Fredholm Grassmannian flow, and for the given linear dispersion relation and corresponding equivalencing group of Fredholm transformations, is unique in the class of odd-polynomial partial differential fields.
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We prove that each member of the non-commutative nonlinear Schrodinger and modified Korteweg--de Vries hierarchy is a Fredholm Grassmannian flow, and for the given linear dispersion relation and corresponding equivalencing group of Fredholm transformations, is unique in the class of odd-polynomial partial differential fields. Thus each member is linearisable and integrable in the sense that time-evolving solutions can be generated by solving a linear Fredholm Marchenko equation, with the scattering data solving the corresponding linear dispersion equation. At each order, each member matches the corresponding non-commutative Lax hierarchy field which thus represent odd-polynomial partial differential fields. We also show that the cubic form for the non-commutative sine--Gordon equation corresponds to the first negative order case in the hierarchy, and establish the rest of the negative order non-commutative hierarchy. To achieve this, we construct an abstract combinatorial algebra, the Poppe skew-algebra, that underlies the hierarchy. This algebra is the non-commutative polynomial algebra over the real line generated by compositions, endowed with the Poppe product -- the product rule for Hankel operators pioneered by Ch. Poppe for classical integrable systems. Establishing the hierarchy members at non-negative orders, involves proving the existence of a `Poppe polynomial' expansion for basic compositions in terms of `linear signature expansions' representing the derivatives of the underlying non-commutative field. The problem boils down to solving a linear algebraic equation for the polynomial expansion coefficients, at each order.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We prove that each member of the non-commutative nonlinear Schrodinger and modified Korteweg--de Vries hierarchy is a Fredholm Grassmannian flow, and for the given linear dispersion relation and corresponding equivalencing group of Fredholm transformations, is unique in the class of odd-polynomial partial differential fields. | result 0.92 | 0.95 | 0.78 | asserted 1.00 |
| s1 | Thus each member is linearisable and integrable in the sense that time-evolving solutions can be generated by solving a linear Fredholm Marchenko equation, with the scattering data solving the corresponding linear dispersion equation. | interpretation 0.72 | 0.79 | 0.24 | asserted 1.00 |
| s2 | At each order, each member matches the corresponding non-commutative Lax hierarchy field which thus represent odd-polynomial partial differential fields. | result 0.98 | 0.70 | 0.12 | asserted 1.00 |
| s3 | We also show that the cubic form for the non-commutative sine--Gordon equation corresponds to the first negative order case in the hierarchy, and establish the rest of the negative order non-commutative hierarchy. | result 0.77 | 0.95 | 0.08 | asserted 1.00 |
| s4 | To achieve this, we construct an abstract combinatorial algebra, the Poppe skew-algebra, that underlies the hierarchy. | method 0.67 | 0.95 | 0.05 | asserted 1.00 |
| s5 | This algebra is the non-commutative polynomial algebra over the real line generated by compositions, endowed with the Poppe product -- the product rule for Hankel operators pioneered by Ch. | method 0.53 | 0.47 | 0.03 | asserted 1.00 |
| s6 | Poppe for classical integrable systems. | background 0.93 | 0.07 | 0.02 | unclear 0.83 |
| s7 | Establishing the hierarchy members at non-negative orders, involves proving the existence of a `Poppe polynomial' expansion for basic compositions in terms of `linear signature expansions' representing the derivatives of the underlying non-commutative field. | method 0.97 | 0.84 | 0.05 | asserted 0.99 |
| s8 | The problem boils down to solving a linear algebraic equation for the polynomial expansion coefficients, at each order. | method 0.98 | 0.80 | 0.05 | asserted 1.00 |
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Original abstract
We prove that each member of the non-commutative nonlinear Schrodinger and modified Korteweg--de Vries hierarchy is a Fredholm Grassmannian flow, and for the given linear dispersion relation and corresponding equivalencing group of Fredholm transformations, is unique in the class of odd-polynomial partial differential fields. Thus each member is linearisable and integrable in the sense that time-evolving solutions can be generated by solving a linear Fredholm Marchenko equation, with the scattering data solving the corresponding linear dispersion equation. At each order, each member matches the corresponding non-commutative Lax hierarchy field which thus represent odd-polynomial partial differential fields. We also show that the cubic form for the non-commutative sine--Gordon equation corresponds to the first negative order case in the hierarchy, and establish the rest of the negative order non-commutative hierarchy. To achieve this, we construct an abstract combinatorial algebra, the Poppe skew-algebra, that underlies the hierarchy. This algebra is the non-commutative polynomial algebra over the real line generated by compositions, endowed with the Poppe product -- the product rule for Hankel operators pioneered by Ch. Poppe for classical integrable systems. Establishing the hierarchy members at non-negative orders, involves proving the existence of a `Poppe polynomial' expansion for basic compositions in terms of `linear signature expansions' representing the derivatives of the underlying non-commutative field. The problem boils down to solving a linear algebraic equation for the polynomial expansion coefficients, at each order.
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What it cites · 16 sentences without a confident label
Builds on 1 paper
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The Pöppe algebra outlined herein represents a generalisation of the Pöppe algebra used in Malham [ref] to derive the non-commutative Korteweg–de Vries hierarchy.
The non-commutative Korteweg--de Vries hierarchy and combinatorial Poppe algebra · Pöppe algebra · 0.99
Compares with 1 paper
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The fourth order non-commutative nonlinear Schrödinger equation for $g=G$, with $ℳ_4=μ_4iℐ$, is given by, [equation] This matches the form given in Malham [ref] and Nijhoff et al. [ref] .
Integrability of local and nonlocal non-commutative fourth order quintic nonlinear Schrodinger equations · Hierarchy examples · 1.00
Cites as background 5 papers
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Again there has been more recent interest in such systems and their solutions, such as Treves [ref] , Hamanaka and Toda [ref] , Degasperis and Lombardo [ref] , Dimakis and Müller–Hoissen [ref] , Carillo and Schiebold [ref] whose results are particularly relevant to those herein, Sooman [ref] , Pelinovsky and Stepanyants [ref] , Buryak and Rossi [ref] , Doikou et al. [ref] , Stylianidis [ref] , Adamopoulou and Papamikos [ref] , Malham [ref] , Gürses and Pekcan [ref] and Ma [ref] .
Integrability of local and nonlocal non-commutative fourth order quintic nonlinear Schrodinger equations · Introduction · 0.89
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Again there has been more recent interest in such systems and their solutions, such as Treves [ref] , Hamanaka and Toda [ref] , Degasperis and Lombardo [ref] , Dimakis and Müller–Hoissen [ref] , Carillo and Schiebold [ref] whose results are particularly relevant to those herein, Sooman [ref] , Pelinovsky and Stepanyants [ref] , Buryak and Rossi [ref] , Doikou et al. [ref] , Stylianidis [ref] , Adamopoulou and Papamikos [ref] , Malham [ref] , Gürses and Pekcan [ref] and Ma [ref] .
Multi-component AKNS systems · Introduction · 0.89
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Recently there has also been a resurgence in this direction as well, see for example, Mulase [ref] , Dupré et al. [ref] , Kasman [ref] , Hamanaka and Toda [ref] , Cafasso [ref] , Cafasso and Wu [ref] and Arthamonov et al. [ref] (also see Beck et al. [ref] and Doikoi et al. [ref] ).
Tau functions, infinite Grassmannians and lattice recurrences · Introduction · 0.99
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