N-order solutions to the Gardner equation in terms of Wronskians

Year 2024, Volume: 7 Issue: 2, 75 - 81, 31.08.2024

Pierre Gaillard

https://doi.org/10.33187/jmsm.1426590

Abstract

$N$-order solutions to the Gardner equation (G) are given in terms of Wronskians of order $N$ depending on $2N$ real parameters. We get solutions expressed with trigonometric or hyperbolic functions.

When one of the parameters goes to $0$, we succeed to get for each positive integer $N$, rational solutions as a quotient of polynomials in $x$ and $t$ depending on $2N$ real parameters. We construct explicit expressions of these rational solutions for the first orders.

Keywords

Gardner equation, Rational Solutions, Wronskians

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