Solitons by Inverting Linear Waves

The linear perspective reveals the complete class of beams whose induced wavegudes are axially uniform, plus their governing equations. These are then solved for any specified nonlinear medium. However, we are mainly interested in illustrative examples which demonstrate existence and novel physics. Surprisingly, many of these can be borrowed directly from the literature of linear waveguides.

The concept is as follows: A nonlinear guided wave is the mode (or modes) of the axially uniform waveguide it induces. Reversing the argument, a mode of a linear waveguide must be a nonlinear guided wave of some nonlinear medium [32]. The particular medium is found simply by inverting the self-consistency relation (1) as is shown pictorially in Fig. 1.

This simple procedure gives closed form expressions necessary to illustrate the novel physics for the various classes of self guided beams, including: families of bright and dark solitons; bistable solitons; dynamic solitons; surface waves; and modes of nonlinear waveguides. It is astonishing that so much can be learned without ever solving a nonlinear equation. We next give some examples.

• Example 1. Bright solitons
• Example 2. Dark solitons
• Example 3. Bistable solitons
• Example 4. Dynamic solitons
• Example 5. Dark-like, bright solitons
• Example 6. Surface waves and nonlinear modes