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Periodic and solitary waves in a Korteweg–de Vries equation with delay
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Qi Qiao, Shuling Yan, Xiang Zhang
Journal:

Proceedings of the Royal Society of Edinburgh. Section A: Mathematics , First View

Published online by Cambridge University Press:

07 September 2023, pp. 1-23
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For a perturbed generalized Korteweg–de Vries equation with a distributed delay, we prove the existence of both periodic and solitary waves by using the geometric singular perturbation theory and the Melnikov method. We further obtain monotonicity and boundedness of the speed of the periodic wave with respect to the total energy of the unperturbed system. Finally, we establish a relation between the wave speed and the wavelength.

Travelling waves for the population genetics model with delay
Guojian Lin, Rong Yuan
Journal:

The ANZIAM Journal / Volume 48 / Issue 1 / July 2006

Published online by Cambridge University Press:

17 February 2009, pp. 57-71
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Under the assumptions that the spatial variable is one dimensional and the distributed delay kernel is the general Gamma distributed delay kernel, when the average delay is small, the existence of travelling wave solutions for the population genetics model with distributed delay is obtained by using the linear chain trick and geometric singular perturbation theory. On the other hand, for the population genetics model with small discrete delay, the existence of travelling wave solutions is obtained by employing a technique which is based on a result concerning the existence of the inertial manifold for small discrete delay equations.

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