Exact conditions for existence of homoclinic orbits in the fifth-order KdV model

We consider homoclinic orbits in the fourth-order equation v(iv) + (1 − ε2)v″ − ε2v = v2 + γ(2vv″ + v′2), where . Numerical computations [CG97, C01] show that homoclinic orbits exist on certain curves γ(ε) in the parameter plane (γ, ε). We study the dependence γ(ε) in the limit ε → 0 and prove that a curve γ(ε) passes through the point (γ0, 0) only if s(γ0) = 0, where s(γ) denotes the Stokes constant for the truncated equation (with ε = 0). The additional condition s′(γ0) ≠ 0 guarantees the existence of a unique curve γ(ε) passing through the point (γ0, 0). Every homoclinic orbit is proved to be single-humped for sufficiently small ε.