Interaction of sine-Gordon kinks with defects: The two-bounce resonance

A model of soliton-defect interactions in the sine-Gordon equations is studied using singular perturbation theory. Melnikov theory is used to derive a critical velocity for strong interactions, which is shown to be exponentially small for weak defects. Matched asymptotic expansions for nearly heteroclinic orbits are constructed for the initial value problem, which are then used to derive analytical formulas for the locations of the well known twoand three-bounce resonance windows, as well as several other phenomena seen in numerical simulations. 1 The two-bounce resonance The two-bounce resonance is a phenomenon displayed by many non-integrable systems in which a solitary wave interacts either with another solitary wave or else with a localized defect in the medium through which it propagates. Fei, Kivshar, and Vázquez study the two-bounce resonance in the sine-Gordon equation perturbed by a localized nonlinear defect [8]. utt − uxx + sinu = ǫδ(x) sinu. (1.1) Kink solitons are initialized propagating (numerically) toward a defect with velocity vi and allowed to interact with the defect. Then one of two things might happen: either the soliton is trapped and comes to rest at the defect location, or else it escapes and propagates away at finite speed vf . (The soliton cannot be destroyed by the interaction because it is defined by its boundary conditions at infinity.) They find that there exists a critical velocity vc. Kink solitons with initial velocity greater than vc pass by the defect. Most solitons with initial speeds below the vc are trapped, remaining at the defect for all times after the interaction time. However, there exist bands of initial velocities, known Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102 Department of Mathematics, Southern Methodist University, Dallas, TX 75275