Existence and Uniqueness of Normalized Multi-peak Solutions for Coupled Nonlinear Schrödinger Systems

Abstract: We consider the following two-component coupled nonlinear Schrödinger (CNLS) system: [ \begin{cases} -Δu +(P(x) + λ) u=μ1 u^3+βu v^2, & \text{in } \mathbb{R}^N,\ -Δv +(Q(x) + λ) v =μ_2 v^3+βvu^2, & \text{in } \mathbb{R}^N \end{cases} ] with the mass constraint $\int{\mathbb{R}^N} (u^2+v^2)\,dx = ρ^2$ for $N=2,3$, where $ρ>0$ is a parameter. By employing the Lyapunov-Schmidt reduction and local Pohozaev identities, we establish the existence and local uniqueness of normalized multi-peak solutions: the result holds for sufficiently small $ρ$ when $N=3$, and for $ρ$ approaching a critical threshold when $N=2$. The main difficulty lies in that the mass constraint involves interactions among all concentration points, while a more refined characterization of such normalized solutions further requires sharp order estimates. In this work, we have discovered some new phenomena that differ from those of solutions without mass constraint and single-peak solutions.

• The paper establishes existence of multi-peak solutions using Lyapunov-Schmidt reduction and singular perturbation methods under L2-mass constraints.
• It provides explicit asymptotic profiles and precise Lagrange multiplier estimates in both N=2 and N=3 cases, ensuring local uniqueness near critical masses.
• The findings have significant implications for modeling two-component Bose-Einstein condensates and spatial multiplexing in nonlinear optical media.

Existence and Uniqueness of Normalized Multi-Peak Solutions for Coupled Nonlinear Schrödinger Systems

Problem Formulation and Context

The paper addresses the existence and local uniqueness of normalized multi-peak solutions in two-component coupled nonlinear Schrödinger (CNLS) systems under $L^2$-mass constraints in $\mathbb{R}^N$, with $N=2,3$. The CNLS system considered models two-component Bose-Einstein condensates (BECs) and nonlinear optical wave propagation, and is given by: $$\begin{cases} - \Delta u +(P(x) + \lambda) u = \mu_1 u^3 + \beta u v^2, \ - \Delta v +(Q(x) + \lambda) v = \mu_2 v^3 + \beta v u^2, \end{cases} \quad \text{in } \mathbb{R}^N,$$ with the mass constraint

$\int_{\mathbb{R}^N}(u^2 + v^2)\, dx = \rho^2.$

Here, $P(x)$ and $Q(x)$ are external potentials, and $\mu_1, \mu_2>0$, $\beta \in \mathbb{R}$ characterize intra- and inter-component nonlinear interactions, respectively. The parameter $\rho$ controls the prescribed $\mathbb{R}^N$0-norm.

Multi-peak normalized solutions are nontrivial critical points characterized both by spatial localization at $\mathbb{R}^N$1 mutually separated points and by adherence to the prescribed mass constraint. Results on such solutions are pertinent to the modeling of multi-component condensates and spatially multiplexed patterns in nonlinear optics.

Main Contributions

Existence and Refined Asymptotics of Multi-Peak Normalized Solutions

Using Lyapunov-Schmidt reduction and local Pohozaev identities, the authors establish the existence of normalized concentration solutions exhibiting $\mathbb{R}^N$2 well-separated peaks centered near non-degenerate coincident critical points of $\mathbb{R}^N$3 and $\mathbb{R}^N$4. The results are stratified by spatial dimension and the regime of $\mathbb{R}^N$5:

• For $\mathbb{R}^N$6, existence holds for sufficiently small $\mathbb{R}^N$7.
• For $\mathbb{R}^N$8 (the $\mathbb{R}^N$9-critical case), existence occurs precisely as $N=2,3$0 from an appropriate side, depending on the sign of a computable coefficient encoding the geometry of the potentials and coupling.

The normalized solutions possess an explicit asymptotic profile: they are closely approximated by a linear superposition of rescalings of ground state solutions to the corresponding limiting system, corrected to higher order by explicit functions solving suitable linearized equations. The locations of the peaks are determined to leading order by the critical points of a weighted combination of the potentials $N=2,3$1 and $N=2,3$2.

Local Uniqueness in the Class of Multi-Peak Normalized Solutions

The work establishes local uniqueness in the following strong sense: for sufficiently small $N=2,3$3 or for $N=2,3$4 sufficiently close to $N=2,3$5 (depending on $N=2,3$6), any two solutions with the prescribed multi-peak form and $N=2,3$7 peaks near the designated critical points must coincide. This is achieved via a delicate contradiction argument, leveraging precise quantitative estimates of the interactions among peaks through local Pohozaev identities and sharp asymptotic characterization of the Lagrange multiplier $N=2,3$8 as a function of $N=2,3$9.

New Phenomena and Technical Insights

A key technical challenge addressed is that, under the constraint, the total mass couples the scales and spatial positions of all peaks—unlike the unconstrained or single-peak case, where peaks may be treated as (almost) independent. The paper provides essential order-sharp expansions for the Lagrange multiplier in dependence on $$\begin{cases} - \Delta u +(P(x) + \lambda) u = \mu_1 u^3 + \beta u v^2, \ - \Delta v +(Q(x) + \lambda) v = \mu_2 v^3 + \beta v u^2, \end{cases} \quad \text{in } \mathbb{R}^N,$$0 and detailed blow-up analysis near criticality, especially in the $$\begin{cases} - \Delta u +(P(x) + \lambda) u = \mu_1 u^3 + \beta u v^2, \ - \Delta v +(Q(x) + \lambda) v = \mu_2 v^3 + \beta v u^2, \end{cases} \quad \text{in } \mathbb{R}^N,$$1-critical case $$\begin{cases} - \Delta u +(P(x) + \lambda) u = \mu_1 u^3 + \beta u v^2, \ - \Delta v +(Q(x) + \lambda) v = \mu_2 v^3 + \beta v u^2, \end{cases} \quad \text{in } \mathbb{R}^N,$$2.

Notably, global uniqueness fails without mass constraint or if peaks are widely separated in parameter space; the constraint introduces a global coupling that underpins uniqueness. Additionally, the refined asymptotics for $$\begin{cases} - \Delta u +(P(x) + \lambda) u = \mu_1 u^3 + \beta u v^2, \ - \Delta v +(Q(x) + \lambda) v = \mu_2 v^3 + \beta v u^2, \end{cases} \quad \text{in } \mathbb{R}^N,$$3 and the solution profiles distinguish the constrained multi-peak setting from its single-peak or unconstrained counterparts.

Technical Approach

The construction of solutions proceeds via:

1. Singular perturbation and blow-up analysis: Reformulation in terms of the singular limit $$\begin{cases} - \Delta u +(P(x) + \lambda) u = \mu_1 u^3 + \beta u v^2, \ - \Delta v +(Q(x) + \lambda) v = \mu_2 v^3 + \beta v u^2, \end{cases} \quad \text{in } \mathbb{R}^N,$$4, leading to a singularly perturbed elliptic system.
2. Profile decomposition: Leading order approximation is a linear combination of $$\begin{cases} - \Delta u +(P(x) + \lambda) u = \mu_1 u^3 + \beta u v^2, \ - \Delta v +(Q(x) + \lambda) v = \mu_2 v^3 + \beta v u^2, \end{cases} \quad \text{in } \mathbb{R}^N,$$5 translates of the unique positive radial ground state of the limiting scalar equation, rescaled to match the mass constraint. Correction terms ($$\begin{cases} - \Delta u +(P(x) + \lambda) u = \mu_1 u^3 + \beta u v^2, \ - \Delta v +(Q(x) + \lambda) v = \mu_2 v^3 + \beta v u^2, \end{cases} \quad \text{in } \mathbb{R}^N,$$6) are obtained by solving associated inhomogeneous linearized systems.
3. Lyapunov-Schmidt reduction: Separation of variables in directions tangent/normal to the constraint manifold; Fredholm theory is used to solve for correction terms in the appropriate orthogonal complement.
4. Local Pohozaev identities: These are systematically employed both to fix the positions of concentration peaks and to unravel the dependence of the Lagrange multiplier and higher-order terms on the problem parameters.
5. Interaction analysis: A careful assessment of inter-peak interactions arising from the mass constraint, which manifest in global quantified nonlinear coupling.
6. Uniqueness via contradiction: Assuming the existence of two distinct normalized multi-peak solutions, an auxiliary normalized difference function is shown, via refined analysis, to converge to zero, contradicting a normalization choice.

Key Results and Estimates

The main results can be summarized as follows.

$$\begin{cases} - \Delta u +(P(x) + \lambda) u = \mu_1 u^3 + \beta u v^2, \ - \Delta v +(Q(x) + \lambda) v = \mu_2 v^3 + \beta v u^2, \end{cases} \quad \text{in } \mathbb{R}^N,$$7	Existence Region for $$\begin{cases} - \Delta u +(P(x) + \lambda) u = \mu_1 u^3 + \beta u v^2, \ - \Delta v +(Q(x) + \lambda) v = \mu_2 v^3 + \beta v u^2, \end{cases} \quad \text{in } \mathbb{R}^N,$$8	Explicit Peak Form	Local Uniqueness?	Constraints
3	$$\begin{cases} - \Delta u +(P(x) + \lambda) u = \mu_1 u^3 + \beta u v^2, \ - \Delta v +(Q(x) + \lambda) v = \mu_2 v^3 + \beta v u^2, \end{cases} \quad \text{in } \mathbb{R}^N,$$9	Superposition of rescaled $\int_{\mathbb{R}^N}(u^2 + v^2)\, dx = \rho^2.$0 at $\int_{\mathbb{R}^N}(u^2 + v^2)\, dx = \rho^2.$1 points	Yes	$\int_{\mathbb{R}^N}(u^2 + v^2)\, dx = \rho^2.$2, nondegeneracy, etc.
2	$\int_{\mathbb{R}^N}(u^2 + v^2)\, dx = \rho^2.$3 near $\int_{\mathbb{R}^N}(u^2 + v^2)\, dx = \rho^2.$4 (depending on sign of $\int_{\mathbb{R}^N}(u^2 + v^2)\, dx = \rho^2.$5)	As above	Yes	As above, plus sum condition (H3)

The Lagrange multiplier $\int_{\mathbb{R}^N}(u^2 + v^2)\, dx = \rho^2.$6 displays dimension-dependent scaling and admits an expansion in terms of $\int_{\mathbb{R}^N}(u^2 + v^2)\, dx = \rho^2.$7:

• For $\int_{\mathbb{R}^N}(u^2 + v^2)\, dx = \rho^2.$8: $\int_{\mathbb{R}^N}(u^2 + v^2)\, dx = \rho^2.$9 as $P(x)$0.
• For $P(x)$1: $P(x)$2 as $P(x)$3, with higher-order corrections explicitly evaluated.

Crucially, uniqueness is conditioned on refined spectral properties of the linearized operator, and the global interaction term in the Pohozaev identity, which links all peaks through the constraint.

Implications and Future Directions

The rigorous construction and uniqueness theory for normalized multi-peak solutions in CNLS systems with explicit higher-order characterization has important consequences:

• Bose-Einstein Condensates: It provides mathematical foundation for the structure and stability of multi-component condensates with prescribed particle number, clarifying the interplay between intra- and inter-component interactions, external trapping, and phase space localization.
• Nonlinear Optics: Multi-peaked stationary states with mass constraints model spatial soliton arrays and multiplexed beams in nonlinear media under global energy (power) conservation.
• Mathematical Theory: The results lay groundwork for spectral stability, impact global bifurcation diagrams, and inform the study of dynamical behavior (e.g., slow motion of peaks, modulation equations).

Possible directions for further research include:

• Stability Analysis: Spectral and nonlinear stability of the constructed solutions, especially under small perturbations of the mass constraint or the potentials.
• Dynamical Emergence: Time-dependent or adiabatic approaches to track the slow formation/interaction of peaks.
• Generalization: Extension to systems with more components, higher dimensions, mass supercritical/subcritical exponents, or spatially nonuniform constraints.
• Singular Limits and Pattern Formation: Detailed study of peak merging, symmetry-breaking bifurcations, and connections to singularly perturbed systems in other physical contexts.

Conclusion

This work rigorously establishes the existence and local uniqueness of normalized multi-peak solutions for $P(x)$4-peak configurations in CNLS systems under mass constraint, providing a detailed quantitative and qualitative description of the profiles, scaling regimes, and the subtle interplay among nonlinearity, external potentials, and normalization. The analysis highlights the nontrivial role of global constraints in coupling peak dynamics and the necessity of precise asymptotics for uniqueness, with direct implications for nonlinear wave physics and PDE theory.