A Direct Construction of Solitary Waves for a Fractional Korteweg-de Vries Equation With an Inhomogeneous Symbol
| Autor: | Yadav, Swati, Xue, Jun |
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| Rok vydání: | 2024 |
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| Druh dokumentu: | Working Paper |
| Popis: | We construct solitary waves for the fractional Korteweg-De Vries type equation $u_t + (\Lambda^{-s}u + u^2)_x = 0$, where $\Lambda^{-s}$ denotes the Bessel potential operator $(1 + |D|^2)^{-\frac{s}{2}}$ for $s \in (0,\infty)$. The approach is to parameterise the known periodic solution curves through the relative wave height. Using a priori estimates, we show that the periodic waves locally uniformly converge to waves with negative tails, which are transformed to the desired branch of solutions. The obtained branch reaches a highest wave, the behavior of which varies with $s$. The work is a generalisation of recent work by Ehrnstr\"om-Nik-Walker, and is as far as we know the first simultaneous construction of small, intermediate and highest solitary waves for the complete family of (inhomogeneous) fractional KdV equations with negative-order dispersive operators. The obtained waves display exponential decay rate as $|x| \to \infty$. Comment: 21 pages |
| Databáze: | arXiv |
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