On the Numerical Approximation to Generalized Ostrovsky Equations: I

A Numerical Method and Computation of Solitary-Wave Solutions
  • Ángel DuránEmail author
Part of the Understanding Complex Systems book series (UCS)


In the present chapter, two numerical procedures to simulate the dynamics of generalized versions of the Ostrovsky equation are presented. First, a numerical method to approximate the corresponding periodic initial-value problem is introduced. The scheme consists of a spatial discretization based on Fourier collocation methods, which is justified by the presence of nonlocal terms. Due to the stiff character of the semidiscretization in space, the time integration is performed with a fourth-order, diagonally implicit Runge-Kutta method, which provides additional theoretical and computational properties. The second point treated in this chapter concerns the solitary-wave solutions of the equations. Their numerical generation is carried out by using a Petviashvili-type method, along with acceleration techniques. The resulting procedure is able to compute both classical and generalized solitary waves in an efficient way. The speed-amplitude relation and the asymptotic behaviour of the waves are studied from the computed profiles.


Generalized Ostrovsky equation Fourier collocation Petviashvili-type methods Solitary waves 



This work was supported by Spanish Ministerio de Economía y Competitividad under the Research Grant MTM2014-54710-P. The author would like to thank Professors V. Dougalis, D. Dutykh and D. Mitsotakis for fruitful discussions and so important suggestions.


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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Applied Mathematics DepartmentUniversity of ValladolidValladolidSpain

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