Winter's Nonlinear Model

• Winter's Nonlinear Model is a one-dimensional quantum system that extends the classical Winter’s model to include nonlinear Schrödinger dynamics with Dirichlet and delta potential conditions.
• It establishes dispersive and Strichartz estimates that underpin the analysis of resonances, survival amplitudes, and well-posedness, linking nonlinear effects to decay rates.
• The model features bifurcation analysis and stability criteria (e.g., Vakhitov–Kolokolov) while highlighting open challenges in applying traditional spectral theory to nonlinear settings.

Winter’s nonlinear model is a one-dimensional quantum system formulated to extend the classical Winter’s model—originally prescribed for linear Schrödinger operators—into the setting of the nonlinear Schrödinger equation (NLS). This model has become a canonical framework for analyzing quantum resonances, survival amplitudes, and local and global well-posedness in the nonlinear context. In particular, it incorporates a Dirichlet wall at the origin and a delta potential at a fixed positive distance, providing a controlled environment for studying boundary-driven and singular-interaction quantum phenomena under nonlinear evolution (Sacchetti, 16 Nov 2025, Sacchetti, 2023).

1. Mathematical Formulation

The nonlinear Winter’s model is posed for wave functions $\psi(x,t)$ supported on the half-line $x \geq 0$ with a Dirichlet boundary at $x=0$ and a point interaction at $x=a>0$ of strength $\alpha\in\mathbb R$. The self-adjoint operator on $L^2(\mathbb R)$ is:

$$H_\alpha = -\frac{d^2}{dx^2} + \beta\,\delta(x) + \alpha\,\delta(x-a), \qquad \beta=+\infty,$$

imposing the boundary and jump conditions:

$$\psi(0) = 0, \quad \psi(a^+)=\psi(a^-), \quad \psi'(a^+)-\psi'(a^-) = \alpha\,\psi(a).$$

The time-dependent nonlinear Schrödinger equation (NLS) is then:

$$\begin{cases} i\partial_t\psi(t,x) = H_\alpha\,\psi(t,x) + \eta\,|\psi(t,x)|^{2\sigma}\psi(t,x), \ \psi(0,x) = \psi_0(x) \in L^2([0,\infty)),\quad \psi_0(0)=0, \end{cases}$$

with $\eta \in \mathbb R$ controlling the nonlinearity (focusing for $\eta<0$, defocusing for $\eta>0$), and $\sigma>0$ the nonlinearity exponent. Conservation laws include mass $\|\psi(t)\|_{L^2}$ and energy

$$\mathcal{E}(\psi) = \langle\psi,H_\alpha\psi\rangle + \frac{\eta}{\sigma+1}\|\psi\|_{L^{2(\sigma+1)}}^{2(\sigma+1)}.$$

(Sacchetti, 16 Nov 2025, Sacchetti, 2023)

2. Dispersive and Strichartz Estimates

For the linearized evolution ($\eta=0$), considering the continuous spectrum subspace (projection $P_c$), the evolution operator satisfies the dispersive decay estimate:

$$\|e^{-itH_\alpha}P_c\varphi\|_{L_x^\infty} \leq C t^{-1/2}\|\varphi\|_{L^1_x}, \quad t>0,$$

with kernel analysis via the free Green’s function and perturbative corrections. This decay matches the free 1D Schrödinger case and underpins the derivation of Strichartz estimates:

$$\|e^{-itH_\alpha}P_c\psi\|_{L^p_tL^q_x} \leq C\|\psi\|_{L^2_x}, \quad \frac{2}{p}+\frac{1}{q}=\frac{1}{2},\ 2\leq p,q\leq\infty.$$

These estimates are critical in establishing well-posedness and nonlinear analysis (Sacchetti, 16 Nov 2025).

3. Well-Posedness, Conservation, and Blow-Up

With initial data $\psi_0 \in H^1([0,\infty))$, $\psi_0(0)=0$, and normalized mass, for any $\sigma>0$ and $\eta\in\mathbb R$, there exists maximal existence time $0<T_{\max}\leq+\infty$ and a unique solution:

$\psi \in C([0,T_\text{max});H^1)$

preserving both $L^2$-norm and energy. The blow-up alternative holds: either $T_\text{max} = +\infty$ (global existence), or $\|\psi(t)\|_{H^1}\to\infty$ as $t\to T_\text{max}^-$ (Sacchetti, 16 Nov 2025). Global existence is ensured under any of:

• $\eta \geq 0$ (defocusing),
• $\eta<0$ and $0<\sigma<2$,
• $\eta<0$, $\sigma=2$, and $\eta$ above a negative threshold,
• $\eta<0$, $\sigma>2$, and $\eta$ above a problem-dependent threshold.

The virial identity, formulated for the variance $I_q(t) = \int_0^\infty (x-q)^2 |\psi(t,x)|^2 dx$, provides functional inequalities for analyzing blow-up mechanisms. Finite-time blow-up is proven for

• negative initial energy $(\mathcal{E}(\psi_0)<0)$,
• power nonlinearity $\sigma>2$,
• and $\eta<\eta_c$ for some threshold $\eta_c<0$, by convexity arguments on $I_a(t)$ (Sacchetti, 16 Nov 2025).

4. Resonances, Survival Amplitudes, and Nonlinear Spectral Analysis

In the linear case ($\eta=0$), resonances are identified as poles in the meromorphic continuation of Green’s functions, corresponding to solutions of $2ik-\lambda + \lambda e^{2ika} = 0$ for the outgoing-wave problem. Nonlinear extension (e.g., with cubic nonlinearity $g|\,\psi|^2\psi$) renders standard spectral theory inapplicable due to nonanalyticity and breakdown of resolvent methods. Instead, "nonlinear resonances" are defined via the real-energy scattering coefficient $S(\Omega)$, determined by matching amplitudes of stationary solutions across the $\delta$–shell at $x=a$ (Sacchetti, 2023).

Sharp peaks in $S(\Omega)$ track resonance-like features; their positions and widths deform continuously as $g$ varies. For moderate $g$, the survival amplitude—$A(t)=\langle \psi_0, \psi(t) \rangle$—displays

$$A(t) = C_\text{res} e^{-i\Omega_\text{res} t} + D t^{-3/2} + \cdots,$$

where $$\Omega_\text{res} = E_\text{res} - i\Gamma_\text{res}/2$$. Repulsive nonlinearity ($g>0$) increases the decay rate ($\operatorname{Im}\Omega_\text{res}$), focusing nonlinearity ($g<0$) reduces it, and beyond a negative threshold $g=\hat g$, an isolated bound state forms and exponential decay ceases (Sacchetti, 2023).

Table: Effects of Nonlinearity on Resonance and Survival Amplitude

Nonlinearity ($g$)	Resonance Peak Shift	Survival Amplitude Decay
$g>0$ (defocusing)	Broader, shifted right	Faster exponential decay
$g<0$, $|g|<|\hat g|$	Sharper, shifted left	Slower decay
$g<0$, $|g|>|\hat g|$	Collapse to bound state	No exponential decay, stationary

(Sacchetti, 2023)

5. Structure and Bifurcation of Stationary Solutions

Stationary solutions ($\psi(t,x) = e^{-i\omega t} \psi(x)$) reduce to a nonlinear eigenvalue problem:

$$H_\alpha\,\psi + \eta|\psi|^{2\sigma}\psi = \omega\psi, \qquad \psi(0)=0.$$

For real $\omega$, $\eta$, and integer $\sigma$, solutions are real up to global phase. Cubic nonlinearity ($\sigma=1$) admits explicit solutions in terms of Jacobi elliptic functions on $(0,a)$ and $(a,\infty)$, subject to matching the Dirichlet and $\delta$–jump conditions. The normalization $\|\psi\|_{L^2}=1$ uniquely selects the nonlinearity parameter $\eta$ for each branch (Sacchetti, 16 Nov 2025).

Numerical analysis reveals a sequence of saddle-node bifurcations: for instance, with $a=1$ and $\alpha=-4$, a primary ground-state branch $\omega_0(\eta)$ continues from the linear ground state at $\eta=0$, and for each negative $\eta_n$, two new branches $\omega_n^\pm(\eta)$ appear for $\eta<\eta_n$ (Sacchetti, 16 Nov 2025, Sacchetti, 2023).

In the cubic case and for the Dirichlet problem at $x=a$ ($\lambda=+\infty$), L$^2$ solutions are given on $(0,a)$ by Jacobi sn (defocusing) or cn (focusing) families, with energies tending to the linear spectrum as $g\to 0$. For finite $\lambda$ and sufficiently large $|g|$, pairs of localized bound states bifurcate at $g = \hat g(\lambda)$ (Sacchetti, 2023).

6. Stability and Spectral Criteria

Stability of standing waves is conjectured following the Vakhitov–Kolokolov (VK) criterion: for a family $\varphi_\omega$ with $\mu(\omega) = \|\varphi_\omega\|_{L^2}$, the sign of $d\mu^2/d\omega$ distinguishes stability:

• $d\mu^2/d\omega > 0$: orbitally stable,
• $d\mu^2/d\omega < 0$: unstable.

For $a=1$, $\alpha=-4$, numerical data confirm that the primary branch $\omega_0(\eta)$ is expected to be orbitally stable ($d\mu^2/d\omega>0$). Bifurcating branches alternate in stability following the sign changes of the VK index (Sacchetti, 16 Nov 2025).

7. Open Problems and Mathematical Challenges

The nonlinear Winter’s model demonstrates several mathematical obstacles in extending standard resonance theory: loss of linear resolvent theory, breakdown of analytic continuation, and the nonanalytic character of the nonlinearity—particularly in attempts to rigorously define or track resonance poles for complex energies. No general spectral-theoretic extension is known for NLS with point interactions. The existing strategy involves tracking resonance analogs via real-energy scattering data and analyzing bifurcations in the stationary regime. Numerical evidence indicates that the link between resonance poles and exponential decay of the survival amplitude, as well as their deformation under nonlinearity, persists in the nonlinear regime (Sacchetti, 2023).

• A. Sacchetti, "Mathematical results for the nonlinear Winter's model," (Sacchetti, 16 Nov 2025)
• A. Sacchetti, "Quantum resonances and analysis of the survival amplitude in the nonlinear Winter's model," (Sacchetti, 2023)