This paper studies how two-dimensional wave packets governed by a nonlinear Schrödinger equation can self-focus and blow up when the nonlinearity is strong. The authors show that, beyond the critical cubic case, families of steady, multi‑peaked waveforms appear. These families ‘‘bifurcate from infinity’’—the individual peaks start infinitely far apart at the critical threshold and come closer together as the nonlinearity exponent increases past that point.

To reach these conclusions the team combined asymptotic analysis and large-scale numerics. They rewrite the equation in stretched variables that follow the focusing (blowup) dynamics and introduce a blowup rate G to track how fast the solution narrows. Varying the nonlinear exponent σ away from the critical value for two dimensions (σ = 1) they computed branches of multi‑peak collapsing solutions and compared them with steady states of the full partial differential equation. They also validated many of their predictions using state‑of‑the‑art finite‑element computations.

At a qualitative level the authors explain the multi‑peak arrangements by an effective ‘‘particle’’ picture. Each localized peak has exponentially decaying tails that interact with the tails of other peaks. Those tail–tail forces, modified by a short-range power‑law factor, compete with a phase‑dependent, on‑site force set by the collapse dynamics. Solving this force balance predicts the spacing of the peaks. In the critical limit the collapse rate G goes to zero, so the inter‑peak distance tends to infinity—hence the phrase ‘‘bifurcations from infinity.’n

The paper also analyzes stability. It finds that multi‑peaked configurations are less stable than the single‑peak collapsing solution. The dominant instability is symmetry breaking: one peak tends to grow relative to the others and wins the competition, producing a single dominant collapse spot. The authors give a systematic description of leading eigenvalues that control these instabilities. For an N‑peak configuration they report a specific counting of eigenvalue groups and how the largest ones scale with the blowup rate G, and they work out the two‑peak case in detail.