Multihumped collapsing solutions in the nonlinear Schrödinger problem: existence, stability, and dynamics

Journal article

In the present work we examine multi-hump solutions of the nonlinear Schrödinger equation in the blowup regime of the one-dimensional model with power law nonlinearity, bearing a suitable exponent of σ > 2. We find that families of such solutions exist for arbitrary pulse numbers, with all of them bifurcating from the critical case of σ = 2. Remarkably, all of them involve “bifurcations from infinity”, i.e., the pulses come inward from an infinite distance as the exponent σ increases past the critical point. The position of the pulses is quantified and the stability of the waveforms is also systematically examined in the so-called “co-exploding frame”. Both the equilibrium distance between the pulse peaks and the point spectrum eigenvalues associated with the multi-hump configurations are obtained as a function of the blowup rate G theoretically, and these findings are supported by detailed numerical computations. Finally, some prototypical dynamical scenarios are explored, and an outlook towards such multi-hump solutions in higher dimensions is provided.

Authors: Chapman, SJ; Kavousanakis, M; Charalampidis, EG; Kevrekidis, IG; Kevrekidis, PG

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Society for Industrial and Applied Mathematics
• Journal: SIAM Journal on Applied Dynamical Systems
• Volume: 25
• Issue: 2
• Pages: 654-694
• Publication date: 2026-04-02
• Acceptance date: 2025-10-22
• DOI: 10.1137/25m1755576
• EISSN: 1536-0040
• ISSN: 1536-0040
• Language: English
• Keywords: nonlinear Schrödinger equation; self-similarity; multi-hump waves; solitary wave; linear stability; WKB