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Perturbed homoclinic solutions in reversible 1:1 resonance vector fields. (English) Zbl 0792.34044
This paper establishes the existence of homoclinic type solutions of smooth, reversible vector fields in \(R^ 4\) with \(1:1\) resonance. More specifically, it studies a four-dimensional vector field, with one pair of double eigenvalues \(\pm iw_ 0\) with two dimensional Jordan blocks, of the form \(dA/dt=iw_ 0A+B+f(\mu,A,\overline A,B, \overline B)\), \(dB/dt=iw_ 0 B+g(\mu,A,\overline A,B, \overline B)\) in \(C^ 2\). The system is reversible in the sense that the symmetry \(S\), defined by \(S(A,B)=(\overline A-\overline B)\), anticommutes with the vector field. It is shown that, in the supercritical case, there are solutions connecting a periodic solution to itself with a phase shift. Moreover, at least two of these solutions have their main amplitude component cancelling in the middle of the orbit. In the subcritical case, it is shown that there is a homoclinic solution tending towards zero at infinity (like a solitary wave). These solutions give new insights into the analysis of nonlinearly resonant surface waves, the study of steady bifurcating solutions in hydrodynamic instability problems taking place in infinitely long cylinders with reflection symmetry, and in the analysis of the (simplified) problem of the flutter of a wing.

MSC:
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
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