Nonautonomous symmetries of the KdV equation and step-like solutions

Адлер, Всеволод Эдуардович

doi:10.1080/14029251.2020.1757236

Cited by 7 publications

(5 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The general theory of such nonlinear special functions, the foundations of which were laid in the work of A.V. Kitaev [20] (simultaneously and independently, one particular case, a higher analogue of the second Painlevé equation, was studied by the first of the authors of this work in [21]) is actively developed today and it finds numerous applications [22]- [50].…”

Section: This Equation Is Reduced To the Heat Equation λmentioning

confidence: 99%

Integrable Abel equation and asymptotics of symmetry solutions of Korteweg-de Vries equation

Suleǐmanov

Shavlukov

2021

Ufimsk. Mat. Zh.

View full text Add to dashboard Cite

We provide a general solution to a first order ordinary differential equation with a rational right-hand side, which arises in constructing asymptotics for large time of simultaneous solutions of the Korteweg-de Vries equation and the stationary part of its higher non-autonomous symmetry. This symmetry is determined by a linear combination of the first higher autonomous symmetry of the Korteweg-de Vries equation and of its classical Galileo symmetry. This general solution depends on an arbitrary parameter. By the implicit function theorem, locally it is determined by the first integral explicitly written in terms of hypergeometric functions. A particular case of the general solution defines self-similar solutions of the Whitham equations, found earlier by G.V. Potemin in 1988. In the well-known works by A.V. Gurevich and L.P. Pitaevsky in early 1970s, it was established that these solutions of the Whitham equations describe the origination in the leading term of non-damping oscillating waves in a wide range of problems with a small dispersion. The result of this work supports once again an empirical law saying that under various passages to the limits, integrable equations can produce only integrable, in certain sense, equations. We propose a general conjecture: integrable ordinary differential equations similar to that considered in the present paper should also arise in describing the asymptotics at large times for other symmetry solutions to evolution equations admitting the application of the inverse scattering transform method.

show abstract

Section: This Equation Is Reduced To the Heat Equation λmentioning

confidence: 99%

Integrable Abel equation and asymptotics of symmetry solutions of Korteweg-de Vries equation

Suleǐmanov

Shavlukov

2021

Ufimsk. Mat. Zh.

View full text Add to dashboard Cite

show abstract

“…We have not touched upon the questions of the analytic properties of the solutions of the constructed reductions. Small-dimensional examples studied in [7,9], as well as similar results on string equations for the KdV equation (see, in particular, [27,28]) allow us to expect that the reductions under study admit special solutions important for physical applications, but their isolation and study remains a very difficult open problem.…”

Section: Darboux-bäcklund Transformationsmentioning

confidence: 86%

“…and this also gives the expression for ũn (in fact, these are the same discrete Miura transformations as in (28), up to a linear change of f n ). Further, calculating the residues at the points α j brings to the relations…”

Section: E Adlermentioning

confidence: 94%

See 1 more Smart Citation

Negative flows and non-autonomous reductions of the Volterra lattice

Adler

2024

Open Communications in Nonlinear Mathematical Physics

View full text Add to dashboard Cite

We study reductions of the Volterra lattice corresponding to stationary equations for the additional, noncommutative subalgebra of symmetries. It is shown that, in the case of general position, such a reduction is equivalent to the stationary equation for a sum of the scaling symmetry and the negative flows, and is written as $(m+1)$-component difference equations of the Painlev\'e type generalizing the dP$_1$ and dP$_{34}$ equations. For these reductions, we present the isomonodromic Lax pairs and derive the B\"acklund transformations which form the $\mathbb{Z}^m$ lattice.

show abstract

“…Remark 8 Recently Adler [51] described a simultaneous solution of the system of evolution equations ( 29), (4) and the system of ODE ( 28), (34). This solution provides an example of an exact solution of the so-called first Gurevich-Pitaevskii problem [52, §8] concerning solutions of the KdV equation ( 4) with step-like initial data possessing different constant limits as x → ±∞.…”

Section: Remarkmentioning

confidence: 99%

Meromorphy of solutions for a wide class of ordinary differential equations of Painlevé type

Domrin,

Shumkin,

Suleimanov

2021

Preprint

View full text Add to dashboard Cite

show abstract

Nonautonomous symmetries of the KdV equation and step-like solutions

Cited by 7 publications

References 22 publications

Integrable Abel equation and asymptotics of symmetry solutions of Korteweg-de Vries equation

Integrable Abel equation and asymptotics of symmetry solutions of Korteweg-de Vries equation

Negative flows and non-autonomous reductions of the Volterra lattice

Meromorphy of solutions for a wide class of ordinary differential equations of Painlevé type

Contact Info

Product

Resources

About