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Nonlinear Schrödinger equation, Bäcklund transformations and Painlevé transcendents

Нелинейное уравнение Шредингера, преобразования Бэклинда и трансцендентные функции Пейнлеве

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Il Nuovo Cimento B (1971-1996)

Summary

In this paper we find the general similarity solution for the nonlinear Schrödinger (NLS) equation in terms of the fourth Painlevé (P 4) transcendents. The application of the Bäcklund transformation (BT) to these transcendents reveals their hidden-simmetry properties. Some explicit families of solutions of theP 4 equation are obtained. We express the general stationary solution for the NLS equation in terms of the Weierstrass ellipticP-functions. In this case the BT is the addition formula for these elliptic functions. Moreover, the BT is used to express the zeros of a fourth-order polynomial via theP-functions, thus recovering a famous formula by Bianchi.

Riassunto

In questo lavoro è espressa la soluzione di similarità più generale per l'equazione di Schrödinger non lineare (NLS) tramite i trascendenti di Painlevé di 4° tipo (P 4). L'applicazione della trasformata di Bäcklund (BT) a questi trascendenti mette in luce le loro proprietà nascoste di simmetria. Si ottengono alcune famiglie di soluzioni esplicite dell'equazioneP 4. Si esprime inoltre la soluzione stazionaria generale dell'equazione NLS tramite le funzioni ellitticheP di Weierstrass. In questo caso la BT diventa la formula di addizione per queste funzioni ellittiche. Infine la BT è usata per esprimere gli zeri di un polinomio di 4° grado tramite le funzioniP, ritrovando così una famosa formula di Bianchi.

Резюме

В этой статье мы находим общее решение для нелинейного уравнения Шредингера в терминах трансцендентных функций Пейнлеве (P 4). Применение преобразования Бэклунда к этим трансцендентным функциям обнаруживает их скрытые свойства симметрии. Получаются некоторые явные семейства решений для уравненияP 4. Мы выражаем общее стационарное решение для нелинейного уравнения Шрединтера в терминах эллиптическихP-дункций Вейерштрасса. В этом случае преобразование Бэклунда представляет дополнительную формулу для этих эллиптических функций. Кроме того, преобразование Бэклунда используется для выражения нулей полинома четвертого порядка черезP-функции, восстанавливая известную формулу Бьянки.

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  1. Istituto di Fisica dell'Università, Lecce

    M. Boiti & F. Pempinelli

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  1. M. Boiti
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  2. F. Pempinelli
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Boiti, M., Pempinelli, F. Nonlinear Schrödinger equation, Bäcklund transformations and Painlevé transcendents. Nuov Cim B 59, 40–58 (1980). https://doi.org/10.1007/BF02739045

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