Hierarchies of the Korteweg–de Vries Equation Related to Complex Expansion and Perturbation
We consider the possibility of constructing a hierarchy of the complex extension of the Korteweg–de Vries equation (cKdV), which under the assumption that the function is real passes into the KdV hierarchy. A hierarchy is understood here as a family of nonlinear partial differential equations with a...
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MDPI AG
2023-04-01
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| Series: | Axioms |
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| Online Access: | https://www.mdpi.com/2075-1680/12/4/371 |
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| author | Tatyana V. Redkina Arthur R. Zakinyan Robert G. Zakinyan Olesya B. Surneva |
| author_facet | Tatyana V. Redkina Arthur R. Zakinyan Robert G. Zakinyan Olesya B. Surneva |
| author_sort | Tatyana V. Redkina |
| collection | DOAJ |
| description | We consider the possibility of constructing a hierarchy of the complex extension of the Korteweg–de Vries equation (cKdV), which under the assumption that the function is real passes into the KdV hierarchy. A hierarchy is understood here as a family of nonlinear partial differential equations with a Lax pair with a common scattering operator. The cKdV hierarchy is obtained by examining the equation on the eigenvalues of the fourth-order Hermitian self-conjugate operator on the invariant transformations of the eigenvector-functions. It is proved that for an operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mover accent="true"><mi>H</mi><mo stretchy="false">^</mo></mover><mi>n</mi></msub></mrow></semantics></math></inline-formula> to transform a solution of the equation on eigenvalues <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfenced><mrow><mover accent="true"><mi>M</mi><mo stretchy="false">^</mo></mover><mo>−</mo><mi>λ</mi><mi>E</mi></mrow></mfenced><mi>V</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> into a solution of the same equation, it is necessary and sufficient that the complex function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mfenced><mrow><mi>x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></mrow></semantics></math></inline-formula> of the operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi>M</mi><mo stretchy="false">^</mo></mover></semantics></math></inline-formula> satisfies special conditions that are the complexifications of the KdV hierarchy equations. The operators <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mover accent="true"><mi>H</mi><mo stretchy="false">^</mo></mover><mi>n</mi></msub></mrow></semantics></math></inline-formula> are constructed as differential operators of order 2<i>n</i> + 1. We also construct a hierarchy of perturbed KdV equations (pKdV) with a special perturbation function, the dynamics of which is described by a linear equation. It is based on the system of operator equations obtained by Bogoyavlensky. Since the elements of the hierarchies are united by a common scattering operator, it remains unchanged in the derivation of the equations. The second differential operator of the Lax pair has increasing odd derivatives while retaining a skew-symmetric form. It is shown that when perturbation tends to zero, all hierarchy equations are converted to higher KdV equations. It is proved that the pKdV hierarchy equations are a necessary and sufficient condition for the solutions of the equation on eigenvalues to have invariant transformations. |
| first_indexed | 2024-03-11T05:14:44Z |
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| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-11T05:14:44Z |
| publishDate | 2023-04-01 |
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| series | Axioms |
| spelling | doaj.art-2787feef046c49e8b5e49687715d40932023-11-17T18:19:26ZengMDPI AGAxioms2075-16802023-04-0112437110.3390/axioms12040371Hierarchies of the Korteweg–de Vries Equation Related to Complex Expansion and PerturbationTatyana V. Redkina0Arthur R. Zakinyan1Robert G. Zakinyan2Olesya B. Surneva3North-Caucasus Center for Mathematical Research, North-Caucasus Federal University, 1 Pushkin Street, 355017 Stavropol, RussiaNorth-Caucasus Center for Mathematical Research, North-Caucasus Federal University, 1 Pushkin Street, 355017 Stavropol, RussiaNorth-Caucasus Center for Mathematical Research, North-Caucasus Federal University, 1 Pushkin Street, 355017 Stavropol, RussiaNorth-Caucasus Center for Mathematical Research, North-Caucasus Federal University, 1 Pushkin Street, 355017 Stavropol, RussiaWe consider the possibility of constructing a hierarchy of the complex extension of the Korteweg–de Vries equation (cKdV), which under the assumption that the function is real passes into the KdV hierarchy. A hierarchy is understood here as a family of nonlinear partial differential equations with a Lax pair with a common scattering operator. The cKdV hierarchy is obtained by examining the equation on the eigenvalues of the fourth-order Hermitian self-conjugate operator on the invariant transformations of the eigenvector-functions. It is proved that for an operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mover accent="true"><mi>H</mi><mo stretchy="false">^</mo></mover><mi>n</mi></msub></mrow></semantics></math></inline-formula> to transform a solution of the equation on eigenvalues <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfenced><mrow><mover accent="true"><mi>M</mi><mo stretchy="false">^</mo></mover><mo>−</mo><mi>λ</mi><mi>E</mi></mrow></mfenced><mi>V</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> into a solution of the same equation, it is necessary and sufficient that the complex function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mfenced><mrow><mi>x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></mrow></semantics></math></inline-formula> of the operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi>M</mi><mo stretchy="false">^</mo></mover></semantics></math></inline-formula> satisfies special conditions that are the complexifications of the KdV hierarchy equations. The operators <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mover accent="true"><mi>H</mi><mo stretchy="false">^</mo></mover><mi>n</mi></msub></mrow></semantics></math></inline-formula> are constructed as differential operators of order 2<i>n</i> + 1. We also construct a hierarchy of perturbed KdV equations (pKdV) with a special perturbation function, the dynamics of which is described by a linear equation. It is based on the system of operator equations obtained by Bogoyavlensky. Since the elements of the hierarchies are united by a common scattering operator, it remains unchanged in the derivation of the equations. The second differential operator of the Lax pair has increasing odd derivatives while retaining a skew-symmetric form. It is shown that when perturbation tends to zero, all hierarchy equations are converted to higher KdV equations. It is proved that the pKdV hierarchy equations are a necessary and sufficient condition for the solutions of the equation on eigenvalues to have invariant transformations.https://www.mdpi.com/2075-1680/12/4/371Lax pairscomplexification of the Korteweg–de Vries equationKorteweg–de Vries hierarchiesintegrable partial differential equationsperturbations of the Korteweg–de Vries equation |
| spellingShingle | Tatyana V. Redkina Arthur R. Zakinyan Robert G. Zakinyan Olesya B. Surneva Hierarchies of the Korteweg–de Vries Equation Related to Complex Expansion and Perturbation Axioms Lax pairs complexification of the Korteweg–de Vries equation Korteweg–de Vries hierarchies integrable partial differential equations perturbations of the Korteweg–de Vries equation |
| title | Hierarchies of the Korteweg–de Vries Equation Related to Complex Expansion and Perturbation |
| title_full | Hierarchies of the Korteweg–de Vries Equation Related to Complex Expansion and Perturbation |
| title_fullStr | Hierarchies of the Korteweg–de Vries Equation Related to Complex Expansion and Perturbation |
| title_full_unstemmed | Hierarchies of the Korteweg–de Vries Equation Related to Complex Expansion and Perturbation |
| title_short | Hierarchies of the Korteweg–de Vries Equation Related to Complex Expansion and Perturbation |
| title_sort | hierarchies of the korteweg de vries equation related to complex expansion and perturbation |
| topic | Lax pairs complexification of the Korteweg–de Vries equation Korteweg–de Vries hierarchies integrable partial differential equations perturbations of the Korteweg–de Vries equation |
| url | https://www.mdpi.com/2075-1680/12/4/371 |
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