Point vortices and polynomials of the Sawada-Kotera and Kaup-Kupershmidt equations

Demina, Maria V.; Kudryashov, Nikolay A.

doi:10.1134/s1560354711060025

Cited by 19 publications

(27 citation statements)

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“…Thus we see that solutions of (32) can be expressed via solutions of the first member of the K 2 hierarchy [25]. This equation admits rational solutions which are expressed via nonlinear special polynomials [17,18]. Let us present some rational solutions of (32).…”

Section: Self-similar Solutionsmentioning

confidence: 99%

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Special solutions of a high-order equation for waves in a liquid with gas bubbles

Kudryashov

Sinelshchikov

2014

Regul. Chaot. Dyn.

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Section: Self-similar Solutionsmentioning

confidence: 99%

“…This equation admits rational solutions that are expressed via nonlinear special polynomials [17,18]. We can also construct rational solutions of (40) using special polynomials from work [17] and Miura transformation (43).…”

Section: Self-similar Solutionsmentioning

confidence: 99%

Special solutions of a high-order equation for waves in a liquid with gas bubbles

Kudryashov

Sinelshchikov

2014

Regul. Chaot. Dyn.

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“…Полиномиальные решения при µ = 1, µ = 2 из-вестны (см. [5][6][7][8]). Уравнение (10) инвариантно относительно преобразования z → αz + β, α = 0, поэтому, без ограничения общности, будем считать, что полином P 2 (z) всегда имеет ноль в начале координат.…”

Section: полиномиальный методunclassified

“…Для описания рациональных решений уравнения Кортевега -де Вриза М. Адлер и Ю. Мозер ввели семейство полиномов, которые, как оказалось, удо-влетворяют уравнению Ткаченко [6]. Связь полиномов, описывающих статические вихревые конфигурации, с некоторыми другими интегрируемыми дифференциаль-ными уравнениями подробно рассматривается в работе [7].…”

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Polynomial method for constructing equilibrium configurations of point vortices in a plane

Demina

Kudryashov

2013

Aut. Control Comp. Sci.

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получена 12 сентября 2012Ключевые слова: точечные вихри, статические конфигурации, равномерно движущиеся конфигурацииРассматривается вопрос построения и классификации статических и рав-номерно движущихся конфигураций точечных вихрей на плоскости при про-извольном выборе интенсивностей вихрей. Дается детальное описание поли-номиального метода, позволяющего находить любую такую конфигурацию. Проводится классификация статических конфигураций для вихрей с интен-сивностями Γ, −µΓ при условии, что µ -целое число, а количество вихрей не превышает десяти. Получены новые конфигурации. ВведениеДвижение M бесконечно тонких прямолинейных параллельных вихревых нитей с интенсивностями Γ 1 , . . ., Γ M в безграничной идеальной жидкости описывается системой уравнений ГельмгольцаЗдесь z k (t) -комплекснозначная функция, определяющая положение точки пересе-чения вихревой нити с перпендикулярной плоскостью в момент времени t. Симво-лом * обозначено комплексное сопряжение, а штрих у знака суммы означает, что необходимо исключить случай m = k. Хорошо известно, что система (1) неинтегри-руема для четырех и более вихрей при произвольном выборе интенсивностей [1]. В связи с этим большое значение имеет задача построения точных решений систе-мы (1), к которым, в частности, относят статические и равномерно движущиеся конфигурации [1][2][3][4].

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“…Roots of the polynomial S(z) give positions of vortices with circulation Γ and roots of the polynomial T (z) give positions of vortices with circulation −µΓ. The case µ = 2 was studied in details in the articles [12,14].…”

Section: Stationary Equilibria Of Point Vorticesmentioning

confidence: 99%

Vortices and polynomials: non-uniqueness of the Adler–Moser polynomials for the Tkachenko equation

Demina¹,

Kudryashov²

2012

J. Phys. A: Math. Theor.

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Stationary and translating relative equilibria of point vortices in the plane are studied. It is shown that stationary equilibria of a system containing point vortices with arbitrary choice of circulations can be described with the help of the Tkachenko equation. It is obtained that the Adler -Moser polynomial are not unique polynomial solutions of the Tkachenko equation. A generalization of the Tkachenko equation to the case of translating relative equilibria is derived. It is shown that the generalization of the Tkachenko equation possesses polynomial solutions with degrees that are not triangular numbers. IntroductionThe problem of finding stationary and relative equilibrium solutions to Helmholtz's equations describing motion of point vortices in the plane has been attracting much attention during recent years [1][2][3][4][5][6][7][8][9]. The so-called "polynomial method" is often used while studying this problem [1]. According to this method one introduces polynomials with roots at vortex positions. For example, it was found by Stieltjes that the generating polynomial of N identical point vortices on a line is essentially the Nth Hermite polynomial. Considering equilibrium of point vortices with equal in absolute value circulations leads to an ordinary differential equation first discovered by Tkachenko [1]. It is well known that the Adler -Moser polynomials give polynomial solutions to the Tkachenko equation. Originally these polynomials arose in 1 the theory of one of the most famous soliton equation, the Korteweg -de Vries equation [10]. This fact provides a remarkable and rather unexpected connection between the dynamics of point vortices and the theory of integrable partial differential equations [11]. Not long ago analogous connections between equilibria of point vortices with circulations Γ, −2Γ and rational solutions of the Sawada -Kotera and the the Kaup -Kupershmidt equations was established [12].In this article we study stationary and translating relative equilibria of multivortex systems with circulations Γ 1 , . . ., Γ N . Our aim is to show that the stationary case can be described with the help of the Tkachenko equation and the translating case is reducible to a generalization of the Tkachenko equation. As a consequence of our results we obtain that the Adler -Moser polynomials are not unique polynomial solutions of the Tkachenko equation.This article is organized as follows. In section 2 we consider stationary equilibria of point vortices with arbitrary choice of circulations. We derive an ordinary differential equation satisfied by generating polynomials of arrangements and transform this equation to the Tkachenko equation. Section 3 is devoted to the translating case. We find an ordinary differential equation satisfied by generating polynomials of the vortices and reduce the resulting equation to the generalization of the Tkachenko equation. In section 4 we study properties of the Tkachenko equation and the generalized Tkachenko equation. Quite unexpectedly it is possible to constr...

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Point vortices and polynomials of the Sawada-Kotera and Kaup-Kupershmidt equations

Cited by 19 publications

References 30 publications

Special solutions of a high-order equation for waves in a liquid with gas bubbles

Special solutions of a high-order equation for waves in a liquid with gas bubbles

Polynomial method for constructing equilibrium configurations of point vortices in a plane

Vortices and polynomials: non-uniqueness of the Adler–Moser polynomials for the Tkachenko equation

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