Transverse super-homoclinic orbit conjecture for coupled nonlinear Schrödinger equations

Let the coupled nonlinear Schrödinger equations be the system given by

$$, and suppose its homoclinic orbits have non-trivial local stable and unstable invariant manifolds that can be globalized. **Transverse super-homoclinic orbit conjecture.** The coupled nonlinear Schrödinger equations given by$$

possess transverse super-homoclinic orbits. The conjecture asserts that the global stable and unstable invariant manifolds intersect transversely along such orbits; the paper presents this as the expected dynamical consequence of the preceding invariant-manifold analysis, with no proof supplied.