Abstract: By combining Fenichel's geometric singular perturbation theory and Melnikov function method, this paper studies the existence of 1-pulse travelling front solutions of a sineGordon equation with slowly varying parameters. Firstly, we get the layer system and the reduced system respectively as well as their global dynamics via the technique of fast-slow separation, and then, we introduce the Melnikov function to determine the transversal intersections between the stable and unstable manifolds of the slow manifold, where we define the so-called Take-off and Touch-down curves. By controlling the Take-off and Touch-down curves to respectively intersect with the stable and unstable manifolds of the saddle points on the slow manifolds transversally, we get the singular heteroclinic orbits with transversality. Correspondingly we get the existence of heteroclinic orbits of the full singularly perturbed system by perturbing such singular heteroclinic orbits. Finally, we consider an example to verify the correctness of the obtained theoretical results.

Key words: sine-Gordon equation, Geometric singular perturbation theory, Melnikov function, 1-Pulse heteroclinic orbit