Soliton formation from a pulse passing the zero-dispersion point in a nonlinear Schr\"odinger equation

We consider in detail the self-trapping of a soliton from a wave pulse that passes from a defocussing region into a focussing one in a spatially inhomogeneous nonlinear waveguide, described by a nonlinear Schrodinger equation in which the dispersion coefficient changes its sign from normal to anomalous. The model has direct applications to dispersion-decreasing nonlinear optical fibers, and to natural waveguides for internal waves in the ocean. It is found that, depending on the (conserved) energy and (nonconserved) mass of the initial pulse, four qualitatively different outcomes of the pulse transformation are possible: decay into radiation; self-trapping into a single soliton; formation of a breather; and formation of a pair of counterpropagating solitons. A corresponding chart is drawn on a parametric plane, which demonstrates some unexpected features. In particular, it is found that any kind of soliton(s) (including the breather and counterpropagating pair) eventually decays into pure radiation with the increase of the energy, the initial mass being kept constant. It is also noteworthy that a virtually direct transition from a single soliton into a pair of symmetric counterpropagating ones seems possible. An explanation for these features is proposed. In two cases when analytical approximations apply, viz., a simple perturbation theory for broad initial pulses, or the variational approximation for narrow ones, comparison with the direct simulations shows reasonable agreement.Comment: 18 pages, 10 figures, 1 table. Phys. Rev. E, in pres

The Modulation of Multiple Phases Leading to the Modified KdV Equation

This paper seeks to derive the modified KdV (mKdV) equation using a novel approach from systems generated from abstract Lagrangians that possess a two-parameter symmetry group. The method to do uses a modified modulation approach, which results in the mKdV emerging with coefficients related to the conservation laws possessed by the original Lagrangian system. Alongside this, an adaptation of the method of Kuramoto is developed, providing a simpler mechanism to determine the coefficients of the nonlinear term. The theory is illustrated using two examples of physical interest, one in stratified hydrodynamics and another using a coupled Nonlinear Schr\"odinger model, to illustrate how the criterion for the mKdV equation to emerge may be assessed and its coefficients generated.Comment: 35 pages, 5 figure

Extreme interfacial waves

Numerical solutions are presented for large-amplitude interfacial waves of extreme form on the interface between two fluids of different densities in the Boussinesq approximation. The flow in the lower fluid is irrotational, but the upper fluid may have constant, nonzero vorticity. Only symmetric waves are calculated. The results suggest limiting wave profiles for which separate portions of the interface touch, forming stagnant zones of one fluid imbedded in the other fluid

Nonlinear interfacial progressive waves near a boundary in a Boussinesq fluid

The behavior of nonlinear progressive waves at the interface between two inviscid fluids in the presence of an upper free boundary is studied as a model of waves on the thermocline. A set of relationships between the integral properties of bounded waves in a general two-fluid model is first developed and the Stokes expansion to third order is derived. The exact free boundary problem for the wave profile is then formulated within the Boussinesq approximation as a nonlinear integral equation, which is solved numerically using two different numerical methods. For finite velocity difference across the two-fluid interface bifurcation of solutions into upper and lower branch wave profiles with quite different properties is obtained. Numerically calculated wave shapes and integral properties show good agreement with third-order Stokes expansion predictions in the weakly nonlinear regime for waves which are not too long. Very long waves were found to exhibit distinct solitary wave-like features