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On the (Non)-Integrability of KdV Hierarchy with Self-consistent Sources
Exactly Solvable and Integrable Systems · Submitted 21 Sep 2011
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim The analysis shows that although many nontrivial solutions of KdV equations with generic self-consistent sources can be obtained by the Inverse Scattering Transform (IST), there are solutions that, in principle, can not be obtained via IST.
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Non-holonomic deformations of integrable equations of the KdV hierarchy are studied by using the expansions over the so-called "squared solutions" (squared eigenfunctions). Such deformations are equivalent to perturbed models with external (self-consistent) sources. In this regard, the KdV6 equation is viewed as a special perturbation of KdV equation. Applying expansions over the symplectic basis of squared eigenfunctions, the integrability properties of the KdV hierarchy with generic self-consistent sources are analyzed. This allows one to formulate a set of conditions on the perturbation terms that preserve the integrability. The perturbation corrections to the scattering data and to the corresponding action-angle variables are studied. The analysis shows that although many nontrivial solutions of KdV equations with generic self-consistent sources can be obtained by the Inverse Scattering Transform (IST), there are solutions that, in principle, can not be obtained via IST. Examples are considered showing the complete integrability of KdV6 with perturbations that preserve the eigenvalues time-independent. In another type of examples the soliton solutions of the perturbed equations are presented where the perturbed eigenvalue depends explicitly on time. Such equations, however in general, are not completely integrable.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Non-holonomic deformations of integrable equations of the KdV hierarchy are studied by using the expansions over the so-called "squared solutions" (squared eigenfunctions). | contribution 0.69 | 0.74 | 0.03 | asserted 0.99 |
| s1 | Such deformations are equivalent to perturbed models with external (self-consistent) sources. | background 0.79 | 0.25 | 0.04 | asserted 1.00 |
| s2 | In this regard, the KdV6 equation is viewed as a special perturbation of KdV equation. | background 0.71 | 0.34 | 0.04 | asserted 0.99 |
| s3 | Applying expansions over the symplectic basis of squared eigenfunctions, the integrability properties of the KdV hierarchy with generic self-consistent sources are analyzed. | method 0.77 | 0.87 | 0.06 | asserted 0.95 |
| s4 | This allows one to formulate a set of conditions on the perturbation terms that preserve the integrability. | result 0.62 | 0.85 | 0.10 | asserted 0.70 |
| s5 | The perturbation corrections to the scattering data and to the corresponding action-angle variables are studied. | contribution 0.89 | 0.85 | 0.04 | asserted 0.89 |
| s6 | The analysis shows that although many nontrivial solutions of KdV equations with generic self-consistent sources can be obtained by the Inverse Scattering Transform (IST), there are solutions that, in principle, can not be obtained via IST. | result 0.98 | 0.86 | 0.69 | asserted 0.94 |
| s7 | Examples are considered showing the complete integrability of KdV6 with perturbations that preserve the eigenvalues time-independent. | result 0.90 | 0.84 | 0.08 | asserted 1.00 |
| s8 | In another type of examples the soliton solutions of the perturbed equations are presented where the perturbed eigenvalue depends explicitly on time. | result 0.83 | 0.84 | 0.06 | asserted 0.99 |
| s9 | Such equations, however in general, are not completely integrable. | result 0.84 | 0.74 | 0.25 | asserted 0.85 |
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Original abstract
Non-holonomic deformations of integrable equations of the KdV hierarchy are studied by using the expansions over the so-called "squared solutions" (squared eigenfunctions). Such deformations are equivalent to perturbed models with external (self-consistent) sources. In this regard, the KdV6 equation is viewed as a special perturbation of KdV equation. Applying expansions over the symplectic basis of squared eigenfunctions, the integrability properties of the KdV hierarchy with generic self-consistent sources are analyzed. This allows one to formulate a set of conditions on the perturbation terms that preserve the integrability. The perturbation corrections to the scattering data and to the corresponding action-angle variables are studied. The analysis shows that although many nontrivial solutions of KdV equations with generic self-consistent sources can be obtained by the Inverse Scattering Transform (IST), there are solutions that, in principle, can not be obtained via IST. Examples are considered showing the complete integrability of KdV6 with perturbations that preserve the eigenvalues time-independent. In another type of examples the soliton solutions of the perturbed equations are presented where the perturbed eigenvalue depends explicitly on time. Such equations, however in general, are not completely integrable.
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How the literature uses it
1 arXiv paper cite this in 1 sentence
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Cites it as background 1 paper
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It is also known that inverse scattering and spectral methods [ref] were applied to generate solutions of equations with self-consistent sources [ref] .
Darboux Transformations for (2+1)-Dimensional Extensions of the KP Hierarchy · Conclusion · 0.89
What it cites · 1 sentence without a confident label
Cites as background 2 papers
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More about the physical relevance of the KdV equation can be found e.g. in [ref] .
Water Waves and Integrability · Preliminaries: Integrability of the Equations of KdV Hierarchy · 1.00
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More about the physical relevance of the KdV equation can be found e.g. in [ref] .
The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations · Preliminaries: Integrability of the Equations of KdV Hierarchy · 1.00
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