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Self-similar asymptotics for solutions to the intermediate long-wave equation

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Abstract

We consider the Cauchy problem for the intermediate long-wave equation

$$\begin{aligned} u_{t}-\partial _{x}u^{2}+\frac{1}{\vartheta }u_{x}+VP\int _{\mathbb {R}}\frac{1}{2\vartheta }\coth \left( \frac{\pi \left( y-x\right) }{2\vartheta }\right) u_{yy}\left( t,y\right) \mathrm{d}y=0, \end{aligned}$$

where \(\vartheta >0\). Our purpose in this paper is to prove the large time asymptotic behavior of solutions under the nonzero mass condition \(\int u_{0}\left( x\right) \mathrm{d}x\ne 0\).

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Acknowledgements

We would like to thank unknown referees for useful suggestions and comments. The work of P.I.N. is partially supported by CONACYT and PAPIIT project IN100616.

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  1. Centro de Ciencias Matemáticas, UNAM Campus Morelia, AP 61-3 (Xangari), CP 58089, Morelia, Michoacán, Mexico

    Fernando Bernal-Vílchis & Pavel I. Naumkin

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Correspondence to Pavel I. Naumkin.

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Bernal-Vílchis, F., Naumkin, P.I. Self-similar asymptotics for solutions to the intermediate long-wave equation. J. Evol. Equ. 19, 729–770 (2019). https://doi.org/10.1007/s00028-019-00498-5

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