Nonlinear differential identities for cnoidal waves |
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| Authors: | Michael Leitner Alice Mikikits‐Leitner |
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| Affiliation: | 1. +49 2. 89 3. 289 4. 11762;5. Heinz Maier‐Leibnitz Zentrum (MLZ), Technische Universit?t München, , 85748 Garching, Germany;6. Zentrum für Mathematik, Technische Universit?t München, , 85748 Garching, Germany |
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| Abstract: | This article presents a family of nonlinear differential identities for the spatially periodic function  , which is essentially the Jacobian elliptic function  with one non‐trivial parameter  . More precisely, we show that this function  fulfills equations of the form ![urn:x-wiley:dummy:mana201300233:equation:mana201300233-math-0005]( "urn:x-wiley:dummy:mana201300233:equation:mana201300233-math-0005") for all  . We give explicit expressions for the coefficients  and  for given s. Moreover, we show that for any s the set of functions  constitutes a basis for  . By virtue of our formulas the problem of finding a periodic solution to any nonlinear wave equation reduces to a problem in the coefficients. A finite ansatz exactly solves the KdV equation (giving the well‐known cnoidal wave solution) and the Kawahara equation. An infinite ansatz is expected to be especially efficient if the equation to be solved can be considered a perturbation of the KdV equation. |
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| Keywords: | Nonlinear wave equations periodic solutions cnoidal waves Korteweg‐de Vries equation Kawahara equation 35Q53 35B10 |
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