Geodesic
On some numerical integration curves for PDE in neighborhood of "butterfly" catastrophe point
Ural mathematical journal, Tome 2 (2016) no. 2, pp. 127-140

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We consider a three-dimensional nonlinear wave equation with the source term smoothly changing over time and space due to a small parameter. The behavior of solutions of this PDE near the typical "butterfly" catastrophe point is studied. In the framework of matched asymptotic expansions method we derive a nonlinear ODE of the second order depending on three parameters to search for the special solution describing the rapid restructuring of the solution of the PDE in a small neighborhood of the catastrophe point, matching with expansion in a more outer layer. Numerical integration curves of the equation for the leading term of the inner asymptotic expansion are obtained.
Keywords: Matched asymptotic expansions, Numerical integration, Butterfly catastrophe, Nonlinear ODE and PDE. 
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     author = {Oleg Yu. Khachay and Pavel A. Nosov},
     title = {On some numerical integration curves for {PDE} in neighborhood of "butterfly" catastrophe point},
     journal = {Ural mathematical journal},
     pages = {127--140},
     publisher = {mathdoc},
     volume = {2},
     number = {2},
     year = {2016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/UMJ_2016_2_2_a10/}
}
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Oleg Yu. Khachay; Pavel A. Nosov. On some numerical integration curves for PDE in neighborhood of "butterfly" catastrophe point. Ural mathematical journal, Tome 2 (2016) no. 2, pp. 127-140. http://geodesic.mathdoc.fr/item/UMJ_2016_2_2_a10/