Stability and instability of the quasilinear Gross--Pitaevskii dark solitons

TL;DR

The paper analyzes the stability of quasilinear Gross–Pitaevskii dark solitons with nonzero boundary data in one dimension. It casts the dynamics in a Hamiltonian framework using hydrodynamic variables and verifies the Grillakis–Shatah–Strauss assumptions to obtain a VK-type stability criterion, with the Hessian spectrum featuring a single negative eigenvalue and an essential spectrum starting at $1-c^2/2$. The authors show that for weak quasilinear effects the dark soliton branch is stable, while stronger quasilinear interactions create a cusp in the energy–momentum diagram that yields a speed-dependent stability dichotomy; they identify a threshold $\kappa_0\approx-3.636$ separating two regimes. These results provide a rigorous description of the stability landscape for quasilinear GP solitons and establish a precise link between the energy–momentum geometry and orbital stability, with potential implications for nonlinear optics and quantum fluids.

Abstract

We study a quasilinear Schrödinger equation with nonzero conditions at infinity. In previous works, we obtained a continuous branch of traveling waves, given by dark solitons indexed by their speed. Neglecting the quasilinear term, one recovers the Gross--Pitaevskii equation, for which the branch of dark solitons is stable. Moreover, Z.~Lin showed that the Vakhitov--Kolokolov~(VK) stability criterion (in terms of the momentum of solitons) holds for general semilinear equations with nonvanishing conditions at infinity. In the quasilinear case, we prove that the VK stability criterion still applies, by generalizing Lin's arguments. Therefore, we deduce that the branch of dark solitons is stable for weak quasilinear interactions. For stronger quasilinear interactions, a cusp appears in the energy-momentum diagram, implying the stability of fast waves and the instability of slow waves.

Figures (2)

• Figure 1: Left panel depicts in grey the energy-momentum diagram of dark solitons $u_{c,\kappa}$ with $\kappa=-50$ it also displays in black the curve $\mathcal{E}_\kappa(\cdot)$. Right panel displays the momentum of the dark soliton $u_{c,\kappa}$ with $\kappa=-50$ as a function of the speed $c$.
• Figure 2: Left panel depicts the energy-momentum diagram of the dark soliton $u_{c,\kappa}$ with $\kappa=-3$. The center panel represents the momentum of the dark soliton $u_{c,\kappa}$ with $\kappa=-3$ as a function of the speed $c$. The right panel displays the value of $\Tilde{c}_\kappa$ depending on $\kappa$, we put $\tilde{c}_\kappa=0$ for $\kappa\geq\kappa_0$.

Theorems & Definitions (19)

• Theorem 1.1: Theorem 1, Proposition 3.8 in delaire2023exotic
• Theorem 1.2: Theorem 1.10, 1.12 and Proposition 6.2 in delaire2023exotic
• Definition 1.3
• Theorem 1.4
• Corollary 1.5
• Remark 2.1
• Theorem 2.2: Theorem 5.1 in benzoniLWP
• Lemma 1
• proof
• Theorem 2.3: Theorem 3 in grillakisshatah