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

Technical Summary: Stability and Instability of Quasilinear Gross–Pitaevskii Dark Solitons

Problem Statement
This work investigates the orbital stability of traveling wave solutions (dark solitons) to a one-dimensional quasilinear Schrödinger equation with nonvanishing conditions at infinity. The equation under consideration is:
$$i\partial_t\Psi + \partial_{xx}\Psi + \Psi(1 -|\Psi|^2) + \kappa\Psi\partial_{xx}(1 -|\Psi|^2) = 0,$$
where $\kappa \in \mathbb{R}$ is a parameter governing the strength of the quasilinear term. When $\kappa = 0$, this reduces to the standard Gross–Pitaevskii (GP) equation, a model widely used in nonlinear optics and quantum hydrodynamics. Previous studies established a continuous branch of dark soliton solutions indexed by their propagation speed $c \in (0, \sqrt{2})$. While the stability of these solitons is well-understood for the semilinear case ($\kappa=0$) via the Vakhitov–Kolokolov (VK) criterion, the impact of the quasilinear term ($\kappa \neq 0$) on stability remained to be rigorously analyzed. Specifically, the paper addresses whether the VK stability criterion, which relates stability to the monotonicity of the momentum with respect to the wave speed, extends to this quasilinear regime.

Methodology
The analysis proceeds by transforming the complex-valued Schrödinger equation into a hydrodynamical system using the Madelung transform $\Psi = \sqrt{1-\eta}e^{i\theta}$, where $\eta$ represents the density deviation and $v = \partial_x \theta$ represents the velocity. This yields a system of equations formally equivalent to the Euler-Korteweg system.

The core methodology relies on the abstract Hamiltonian framework developed by Grillakis, Shatah, and Strauss (GSS) and its subsequent generalization by Z. Lin for equations with nonvanishing boundary conditions. The approach involves:

1. Hamiltonian Formulation: Casting the hydrodynamical system as $\partial_t(\eta, v) = J E'(\eta, v)$, where $E$ is the energy functional and $P$ is the momentum. The traveling wave solutions are identified as critical points of the action functional $E - cP$.
2. Spectral Analysis: Performing a detailed spectral analysis of the Hessian operator $H_c = E'' - cP''$ linearized around the dark soliton solution. The authors verify the three key assumptions required for the GSS stability theorem:
   • Local well-posedness of the initial value problem in appropriate Sobolev spaces ($H^s \times H^{s-1}$ with $s > 2.5$).
   • Existence and smoothness of the branch of bound states (dark solitons).
   • Spectral decomposition of the Hessian, specifically identifying the number of negative eigenvalues and the kernel structure.
3. Adaptation of Lin's Arguments: Since the symplectic operator $J$ in this hydrodynamical setting is not surjective (a technical hurdle preventing the direct application of the instability converse in the original GSS framework), the authors adapt the construction of an energy-decreasing direction used by Lin. This allows for a rigorous proof of instability when the VK criterion is violated.
4. Metric Equivalence: The paper establishes the equivalence between the "hydrodynamical distance" ($d_{hy}$) and the "natural energy distance" ($d_X$) in the neighborhood of the soliton, ensuring that stability results derived in the hydrodynamical variables translate to the original energy space.

Key Contributions and Results
The primary contribution is the rigorous justification of the VK stability criterion for the quasilinear Gross–Pitaevskii equation. The main results are:

• Generalization of the VK Criterion: The paper proves that the orbital stability of the dark soliton $u_{c,\kappa}$ is determined by the sign of the derivative of the momentum $P$ with respect to the speed $c$. Specifically:
  • If $\frac{dP}{dc} < 0$, the soliton is orbitally stable.
  • If $\frac{dP}{dc} > 0$ (and assuming no finite-time blow-up occurs), the soliton is orbitally unstable.
• Stability Regimes based on $\kappa$: By analyzing the explicit dependence of the momentum on $c$ and $\kappa$, the authors derive two distinct stability regimes:
  1. Weak Quasilinear Interactions ($\kappa \geq \kappa_0$): For $\kappa$ in the range $[\kappa_0, 1/2)$, where $\kappa_0 \approx -3.636$, the entire branch of dark solitons ($c \in (0, \sqrt{2})$) is orbitally stable. This recovers the stability behavior of the standard Gross–Pitaevskii equation ($\kappa=0$).
  2. Strong Quasilinear Interactions ($\kappa < \kappa_0$): For sufficiently negative $\kappa$, the momentum-speed curve develops a cusp. In this regime, there exists a critical speed $\tilde{c}_\kappa$ such that:
     • Slow waves ($c < \tilde{c}_\kappa$) are orbitally unstable.
     • Fast waves ($c > \tilde{c}_\kappa$) are orbitally stable.
• Spectral Properties: The work provides a detailed spectral analysis of the linearized operator $L_c$, confirming the existence of exactly one negative eigenvalue and characterizing the essential spectrum, which is necessary to apply the abstract stability theorems.

Significance and Claims
The paper claims to extend the theoretical understanding of soliton stability from semilinear to quasilinear dispersive equations with nonvanishing boundary conditions. By generalizing Lin's arguments, the authors demonstrate that the VK criterion remains a robust predictor of stability even in the presence of quasilinear terms, provided the interaction strength is not too large.

The significance lies in the identification of a transition in stability behavior driven by the parameter $\kappa$. The existence of a "cusp" in the energy-momentum diagram for strong quasilinear interactions implies that while fast waves remain stable, the introduction of strong nonlocal effects can destabilize slow-moving dark solitons. This contrasts with the semilinear case where the entire branch is stable. The authors note that their analysis is based on the assumption of global existence (no blow-up) for the instability proof, a condition consistent with the local well-posedness results available for this class of equations. The work does not propose new experimental applications but rather provides a rigorous mathematical foundation for understanding the stability properties of these specific hydrodynamical models.