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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2407.02717 |
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| _version_ | 1866909240590860288 |
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| author | Yadav, Swati Xue, Jun |
| author_facet | Yadav, Swati Xue, Jun |
| contents | We construct solitary waves for the fractional Korteweg-De Vries type equation $u_t + (Λ^{-s}u + u^2)_x = 0$, where $Λ^{-s}$ 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 Ehrnström-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 $|x| \to \infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_02717 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Direct Construction of Solitary Waves for a Fractional Korteweg-de Vries Equation With an Inhomogeneous Symbol Yadav, Swati Xue, Jun Analysis of PDEs 76B25, 35C07, 76B03 We construct solitary waves for the fractional Korteweg-De Vries type equation $u_t + (Λ^{-s}u + u^2)_x = 0$, where $Λ^{-s}$ 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 Ehrnström-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 $|x| \to \infty$. |
| title | A Direct Construction of Solitary Waves for a Fractional Korteweg-de Vries Equation With an Inhomogeneous Symbol |
| topic | Analysis of PDEs 76B25, 35C07, 76B03 |
| url | https://arxiv.org/abs/2407.02717 |