Solutions with one dimensional concentration for a two dimensional Gross-Pitaevskii model with general potential

1. Problem Statement

The paper investigates standing wave solutions to the two-dimensional Gross-Pitaevskii (GP) equation with an attractive interaction and an external trap potential $W(y)$, subject to a fixed $L^2$-mass constraint. The governing equations are:
$$\begin{cases} i\partial_t \Psi = -\Delta \Psi + W(y)\Psi - a|\Psi|^2\Psi, & (t, y) \in \mathbb{R} \times \mathbb{R}^2, \\ \int_{\mathbb{R}^2} |\Psi|^2 dy = 1. \end{cases}$$
The authors seek excited state solutions of the form $\Psi(t, y) = \tilde{u}(y)e^{-i\lambda t}$, where $\lambda$ is a frequency parameter. As the interaction strength $a$ approaches a critical value $a^*$, the frequency $\lambda \to -\infty$. By rescaling, the problem is transformed into a singular perturbation problem for a small parameter $\epsilon = \sqrt{-1/\lambda}$:
$$-\epsilon^2 \Delta u + [1 + \epsilon^2 W(y)]u - |u|^2 u = 0, \quad \int_{\mathbb{R}^2} |u|^2 dy = a\epsilon^2.$$
Key Challenge: Unlike previous works that assumed radial symmetry (reducing the problem to ODEs), this paper addresses the case where the potential $W$ is non-symmetric. The goal is to construct solutions that concentrate along a closed smooth curve $\Gamma_\epsilon$ (a 1D submanifold in 2D space) as $\epsilon \to 0$, where the length of the curve is not uniformly bounded.

2. Methodology

The authors employ a sophisticated infinite-dimensional Lyapunov-Schmidt reduction method, adapted from the work of Del Pino, Kowalczyk, and Wei, but significantly extended to handle the mass constraint and non-uniformly bounded curves.

A. Necessary Conditions (Theoretical Analysis)

Before constructing solutions, the authors derive necessary conditions for the existence of such concentrating solutions:

1. Stationary Condition: The curve $\Gamma_\epsilon$ must be a critical point of a weighted length functional involving the potential $W$. Specifically, the curvature $\kappa$ and the normal derivative of the potential $W_t$ must satisfy a specific balance relation:
   $$\kappa(\theta)[1 + \epsilon^2 W(0, \theta)] + \frac{3}{2}\epsilon^2 W_t(0, \theta) = 0.$$
2. Mass Constraint: The total mass constraint implies that the interaction strength $a$ must tend to infinity ($a \to +\infty$) as $\epsilon \to 0$.

B. Construction Strategy

The construction involves several rigorous steps:

1. Fermi Coordinates & Local Approximation: The problem is reformulated in Fermi coordinates $(t, \theta)$ near the curve $\Gamma_\epsilon$. A local approximate solution is constructed using the ground state profile $U$ of the 1D GP equation, modulated by the potential and the curve's geometry.
2. Correction Terms: To improve the approximation, the authors introduce correction terms involving two free parameters:
   • $f(\tau)$: A function describing the deformation of the curve.
   • $e(\tau)$: A function related to the amplitude modulation.
     These corrections eliminate leading-order error terms in the expansion.
3. Gluing Procedure: A global approximate solution is constructed by "gluing" the inner solution (near the curve) with the outer solution (decaying away from the curve) using cut-off functions.
4. Projected Problem: The error of the approximate solution is projected onto the kernel of the linearized operator (spanned by $U'$ and $Z$). This leads to a system of coupled differential equations for the parameters $f$ and $e$.
5. Solving the Reduced System: The reduced system is a nonlinear coupled system of second-order ODEs. The authors prove its solvability using:
   • Non-degeneracy conditions: Ensuring the linearized operator is invertible.
   • Gap conditions: Avoiding resonances between the eigenvalues of the linearized operator and the periodicity of the curve.
   • Contraction Mapping & Schauder Fixed Point Theorems: To handle the nonlinearities and compact operators.
6. Parameter Selection: Finally, the parameter $\epsilon$ (and thus $\lambda$) is chosen to satisfy the global mass constraint $\int |u|^2 = a\epsilon^2$.

3. Key Contributions

1. Removal of Symmetry Assumptions: This is the first rigorous construction of solutions concentrating on curves for the 2D GP equation with a general, non-symmetric potential. Previous results (e.g., by Guo et al.) relied heavily on radial symmetry.
2. Non-Uniformly Bounded Curves: The paper addresses the case where the length of the concentrating curve $\Gamma_\epsilon$ is not uniformly bounded as $\epsilon \to 0$. This introduces significant technical difficulties regarding the spectrum of the linearized operator and the behavior of the weighted length functional.
3. Necessary Conditions for General Potentials: The authors provide a rigorous proof of the necessary conditions for concentration on curves in the non-radial case, confirming a conjecture related to the Ambrosetti-Malchiodi-Ni framework.
4. Handling Mass Constraints: The work pioneers the application of infinite-dimensional Lyapunov-Schmidt reduction to nonlinear elliptic problems with a mass constraint, showing how the constraint dictates the scaling of the interaction strength $a$.

4. Main Results

• Theorem 2.2 (Necessary Conditions): If a solution concentrates on a smooth closed curve $\Gamma_\epsilon$, then:
  • The length of the curve is bounded below ($\ell \ge \ell_0$).
  • The interaction strength $a \to +\infty$.
  • The curve must satisfy the stationary condition relating curvature and the potential gradient (Equation 2.17).
• Theorem 2.5 & 2.6 (Existence): Under specific assumptions on the potential $W$ (smoothness, positivity, decay of derivatives) and the curve $\Gamma_\epsilon$ (stationarity, non-degeneracy, and a gap condition to avoid resonance), there exists a sequence of parameters $a \to \infty$ such that the GP equation admits a positive solution concentrating along $\Gamma_\epsilon$.
  • The solution has the local profile: $u_\epsilon(y) \approx \sqrt{1+\epsilon^2 W} \, U\left(\frac{\sqrt{1+\epsilon^2 W} \, t}{\epsilon}\right)$.
  • The solution decays exponentially away from the curve.

5. Significance

• Mathematical Physics: The results deepen the understanding of Bose-Einstein Condensates (BECs) with attractive interactions in non-symmetric traps. It explains how excited states can form ring-like structures (vortices or solitons) that are not necessarily circular, depending on the trap geometry.
• Analytical Advancement: The paper overcomes the "resonance" difficulties inherent in singular perturbation problems on manifolds with unbounded length. The introduction of new weighted norms and the careful handling of the gap conditions provide a robust framework for future studies on concentration phenomena in higher dimensions or with more complex constraints.
• Conjecture Resolution: It partially resolves the open problem regarding the necessary conditions for concentration on submanifolds raised by Ambrosetti, Malchiodi, and Ni, extending their results from radial to general settings.

In summary, this paper provides a rigorous existence theory for high-dimensional concentration phenomena in the Gross-Pitaevskii model, bridging the gap between symmetric and general potential scenarios while addressing the complexities introduced by mass constraints and unbounded geometry.