Abstract
Ginzburg–Landau equation plays a very important role in a wide range of physical contexts. Much attention had been given to explore the dynamics of the waves in Ginzburg–Landau equations as well as its various coupled reaction–diffusion systems. In this article, dynamical system techniques such as the methods of geometric singular perturbation theory and Melnikov function are combined to show the persistence (existence) and bifurcations of traveling fronts in a real supercritical Ginzburg–Landau equation coupled by a slow diffusion mode. It is shown that heteroclinic saddle-node bifurcation occurs under certain parametric values such that 1, 2 and even 3 heteroclinic solutions may exist. The critical manifolds of the parameters corresponding to the heteroclinic saddle-node bifurcations are explicitly determined.
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Acknowledgements
Supported by the National Natural Science Foundations of China and Fujian Province (Nos. 11771082, 11401229, 2015J01004), New Century Excellent Talents in Fujian Province, and the Nonlinear Analysis Innovation Team (IRTL 1206) funded by FJNU.
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Tu, M., Shen, J. & Zhou, Z. Traveling Fronts of a Real Supercritical Ginzburg–Landau Equation Coupled by a Slow Diffusion. Qual. Theory Dyn. Syst. 17, 29–48 (2018). https://doi.org/10.1007/s12346-017-0264-x
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DOI: https://doi.org/10.1007/s12346-017-0264-x