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EXISTENCE OF TRAVELLING WAVES IN THE FRACTIONAL BURGERS EQUATION

Part of: Partial differential equations Miscellaneous topics - Partial differential equations Representations of solutions Singular integral equations

Published online by Cambridge University Press:  20 December 2017

ADAM CHMAJ Open the ORCID record for ADAM CHMAJ [Opens in a new window]
ADAM CHMAJ*
Affiliation:
Freelance, Warsaw, Poland email [email protected]
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Abstract

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We construct travelling waves in the Burgers equation with the fractional Laplacian $(D^{2})^{\unicode[STIX]{x1D6FC}}$, $\unicode[STIX]{x1D6FC}\in (1/2,1)$. This is done by first constructing odd solutions $u_{\unicode[STIX]{x1D700}}$ of $uu^{\prime }=K_{\unicode[STIX]{x1D700}_{1}}\ast u-k_{\unicode[STIX]{x1D700}_{1}}u+\unicode[STIX]{x1D700}_{2}u^{\prime \prime }$, $u(-\infty )=u_{c}>0$, with $K_{\unicode[STIX]{x1D700}_{1}}\ast u-k_{\unicode[STIX]{x1D700}_{1}}u$ nonsingular, and then passing to the limit $\unicode[STIX]{x1D700}_{1},\unicode[STIX]{x1D700}_{2}\rightarrow 0$, to give $K_{\unicode[STIX]{x1D700}_{1}}\ast u_{\unicode[STIX]{x1D700}}-k_{\unicode[STIX]{x1D700}_{1}}u_{\unicode[STIX]{x1D700}}\rightarrow (D^{2})^{\unicode[STIX]{x1D6FC}}u_{0}$ pointwise. The proof relies on operator splitting.

Keywords

singular integral equationtravelling waves

MSC classification

Primary: 45E10: Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
Secondary: 35A01: Existence problems 35C07: Traveling wave solutions 35R11: Fractional partial differential equations
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 97 , Issue 1 , February 2018 , pp. 102 - 109
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

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