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- Title
Hierarchies of the Korteweg–de Vries Equation Related to Complex Expansion and Perturbation.
- Authors
Redkina, Tatyana V.; Zakinyan, Arthur R.; Zakinyan, Robert G.; Surneva, Olesya B.
- Abstract
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 H ^ n to transform a solution of the equation on eigenvalues M ^ − λ E V = 0 into a solution of the same equation, it is necessary and sufficient that the complex function u x , t of the operator M ^ satisfies special conditions that are the complexifications of the KdV hierarchy equations. The operators H ^ n are constructed as differential operators of order 2n + 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.
- Subjects
KORTEWEG-de Vries equation; NONLINEAR differential equations; PARTIAL differential equations; DIFFERENTIAL operators; OPERATOR equations; HERMITIAN forms; LAX pair
- Publication
Axioms (2075-1680), 2023, Vol 12, Issue 4, p371
- ISSN
2075-1680
- Publication type
Article
- DOI
10.3390/axioms12040371