Interactions Between Solitons and Other Nonlinear Schr"odinger Waves

The Nonlinear Schr"odinger (NLS) equation is widely used in everywhere of natural science. Various nonlinear excitations of the NLS equation have been found by many methods. However, except for the soliton-soliton interactions, it is very difficult to find interaction solutions between different types of nonlinear excitations. In this paper, three very simple and powerful methods, the symmetry reduction method, the truncated Painlev'e analysis and the generalized tanh function expansion approach, are further developed to find interaction solutions between solitons and other types of NLS waves. Especially, the soliton-cnoidal wave interaction solutions are explicitly studied in terms of the Jacobi elliptic functions and the third type of incomplete elliptic integrals. In addition to the new method and new solutions of the NLS equation, the results can unearth some new physics. The solitons may be decelerated/accelerated through the interactions of soliton with background waves which may be utilized to study tsunami waves and fiber soliton communications; the static/moving optical lattices may be automatically excited in all mediums described by the NLS systems; solitons elastically interact with non-soliton background waves, and the elastic interaction property with only phase shifts provides a new mechanism to produce a controllable routing switch that is applicable in optical information and optical communications.

💡 Research Summary

The paper addresses a long‑standing difficulty in nonlinear wave theory: obtaining explicit interaction solutions between solitons and other types of excitations of the nonlinear Schrödinger (NLS) equation. While multi‑soliton solutions are well known, analytical expressions for a soliton interacting with non‑solitonic backgrounds such as cnoidal waves, Painlevé‑type waves, or other periodic structures have been scarce. To overcome this, the authors develop three complementary techniques that are both simple to implement and powerful in scope.

First, the NLS equation is recast as a special case of the Ablowitz‑Kaup‑Newell‑Segur (AKNS) system. The authors review the infinite hierarchy of local symmetries (K‑series) and non‑local symmetries (τ‑series, square‑eigenfunction symmetries). By exploiting the λ‑dependence of the square‑eigenfunction symmetry N₀, they generate an infinite family of non‑local symmetries Nₙ either through the inverse recursion operator or by repeated differentiation with respect to the spectral parameter λ.

Because non‑local symmetries cannot be directly used for reduction, the authors introduce a “localization” procedure. They augment the AKNS system with the Lax pair and auxiliary spectral functions φ₁, φ₂, together with a scalar field φ satisfying φₓ = φ₁φ₂/√b and φₜ = (i/2)(qφ₂² – pφ₁²) – 2√b λ φ₁φ₂. Solving an initial‑value problem for the infinitesimal transformation generated by N₀ yields a finite auto‑Bäcklund transformation (ABT): p′ = p + ε φ₁² – ε φ, q′ = q + ε φ₂² – ε φ, φ₁′ = φ₁ – ε φ, φ₂′ = φ₂ – ε φ, φ′ = φ – ε φ. This ABT provides a concrete finite map associated with the non‑local symmetry.

Next, the authors seek group‑invariant solutions that are simultaneously invariant under a selected set of local symmetries (K₀, K₁, K₂, τ₀, τ₁) and the non‑local symmetry N₀. By imposing the invariant condition σₙₗ = 0 on the enlarged system, they introduce new similarity variables ξ = x + α t and τ = β t + γ. The resulting invariant solutions have the generic structure φ(ξ,τ) = C₁ tanh