Nonlinear Schrödinger Energy

• Nonlinear Schrödinger energy is defined as the variational formulation coupling kinetic, potential, and nonlinear interaction terms to analyze the existence and stability of solutions.
• It underpins ground state theory by serving as a conserved Hamiltonian that drives minimization under mass constraints and governs bifurcation phenomena.
• Structure-preserving numerical methods, such as Crank–Nicolson and discontinuous Galerkin schemes, are employed to accurately conserve energy and ensure spectral stability.

The nonlinear Schrödinger energy is a central object in the mathematical analysis and numerical computation of nonlinear Schrödinger equations (@@@@1@@@@), governing the existence, stability, and dynamics of solutions. In these models, the energy functional serves as a Lyapunov (or Hamiltonian) functional, often controlling global dynamics, ground-state structures, bifurcation phenomena, and spectral stability. The precise form of the nonlinear Schrödinger energy depends on the nonlinearity, the presence of external potentials, and the dimension, but typically couples kinetic, potential, and nonlinear terms in a variational framework.

1. Definition and Variational Structure

The standard NLSE in a domain Ω ⊂ ℝ^d for a wave function $\psi: \mathbb{R}^d \to \mathbb{C}$ with power-type nonlinearity is given by

$$i\partial_t\psi = -\frac{1}{2}\Delta\psi + V(x)\psi + \lambda |\psi|^{2\sigma}\psi,$$

where $V(x)$ is an external potential, $\lambda$ is the interaction strength, and $\sigma>0$ is the nonlinearity index (Besse et al., 2018, Ruan, 2017).

The associated energy (Hamiltonian) functional, conserved under the NLSE flow, is

$$E[\psi] = \int_{\Omega} \Bigg[ \frac{1}{2}|\nabla\psi|^2 + V(x)|\psi|^2 + \frac{\lambda}{\sigma+1}|\psi|^{2\sigma+2} \Bigg] dx,$$

with suitable adaptations for nonlocal or more exotic nonlinearities (Besse et al., 2018, Ruan, 2017, Lin et al., 2016, Liew et al., 2009). In the case of time-harmonic or standing-wave solutions, this energy is minimized or constrained under a fixed $L^2$ mass, yielding the ground state or nonlinear bound states (Garrisi et al., 2023, Dovetta et al., 2021).

2. Energy Functionals for Non-Standard Nonlinearities

In addition to pure-power cases, the nonlinear Schrödinger energy appears in forms dictated by the physical context or mathematical generalization:

• Algebraic Nonlinearities: For models where the nonlinearity is defined via an algebraic formula $G(s)$, e.g., $G(s) = -\frac{1}{2}s^2 V(s)$ where $V$ satisfies $s^2 = a V^3 + b V^2 + c V$, the energy for $u \in H^1(\mathbb{R}; \mathbb{C})$ becomes

$$E(u) = \frac{1}{2} \int |u'|^2 dx + \int G(|u|) dx,$$

and is used in constrained minimizations with fixed mass (Garrisi et al., 2023).

• Saturable and Square-root Nonlinearities: In nonlinear optics, $f(s) = 1 - 1/\sqrt{1+s}$ (square-root) or $f(s)=1 - 1/(1+s)$ (saturable), and the potential $F(s)$ is obtained by integrating $f$, leading to $E[u] = K[u] + P_\Gamma[u]$ with analytic formulas for $F$ (Lin et al., 2016).
• q-Deformed and Information-Theoretic Nonlinearities: When the energy is constructed with information-theoretic measures (e.g., the Kullback–Leibler divergence), one arrives at non-polynomial, $q$-deformed energy functionals of the form

$$E_q[\psi] = \int \left[ \frac{\hbar^2}{2m} |\partial_x \psi|^2 + V(x) |\psi|^2 \right] dx - \frac{\lambda}{q\eta^4} \int p(x) \ln_q \left(\frac{p(x)}{(1-\eta)p(x)+\eta p(x+\eta L)}\right) dx,$$

with $\ln_q$ the $q$-logarithm (Liew et al., 2009).

• Nonlinear, Non-signaling Models: For certain nonlinear extensions of NLSE enforcing non-signaling or collapse effects, the “energy” modifies the kinetic component, e.g. with flipped sign above a critical mass, but may not always generate a strictly conserved Hamiltonian (Geszti, 2024).

3. Role in Ground State Theory and Bifurcations

The nonlinear Schrödinger energy underpins ground-state existence, uniqueness, and bifurcation phenomena:

• Variational Minimization: The ground state is the minimizer of $E[\psi]$ under a mass constraint (e.g., $\|\psi\|_2^2 = 1$) (Besse et al., 2018, Ruan, 2017, Garrisi et al., 2023). Associated Lagrange multipliers yield the chemical potential or nonlinear eigenvalue.
• Multiplicity and Degeneracy: For certain algebraic nonlinearities, the energy landscape exhibits degenerate minima or multiplicity of positive, radial minimizers with equal mass and energy. The structure of the function $\lambda(\omega)$ (the mass as a function of frequency) dictates uniqueness versus multiplicity, through spectral/variational identities (Garrisi et al., 2023).
• Energy Asymptotics: In regimes of strong or weak nonlinearity, asymptotic expansions and the Thomas–Fermi approximation provide analytic formulas for the ground state energy, e.g., in harmonic traps or boxes (Ruan, 2017). For high nonlinearity, logarithmic corrections, such as $E_\Gamma \sim -\Gamma + T \ln \Gamma + O(1)$, can emerge (Lin et al., 2016).
• Scattering and Energy Thresholds: In mass-supercritical regimes and focusing/defocusing scenarios, the conserved energy, together with the ground-state threshold, governs global well-posedness, blowup, or scattering dynamics (Guo et al., 2019).

4. Spectral Stability and Orbital Properties

The second variation (Hessian) of the nonlinear Schrödinger energy at a minimizer determines spectral (linearized) stability:

• Nondegeneracy and Hessian Quadratic Form: The nondegeneracy of a minimizer is equivalent to the positivity of the quadratic form $Q_R(h)$ on the tangent space of mass-constraint, signaling isolated, robust ground states. The derivative $\lambda'(\omega)$ governs spectral stability via the Vakhitov–Kolokolov (VK) criterion (Garrisi et al., 2023).
• Orbital Stability: Sets of minimizers up to phase and translation (the orbit $\mathcal{G}_\lambda$) are orbitally stable if they minimize energy under mass constraint and the Hessian is positive. The VK criterion relates the sign of $dM/d\omega$ to orbital stability (Garrisi et al., 2023).
• Legendre-Fenchel Duality: The energy ground states constrained by mass and the action ground states constrained by frequency are dual via Legendre-Fenchel transformation; differentiability and convexity breakdowns signal bifurcations or multiplicity (Dovetta et al., 2021).

5. Numerical Methods: Structure-Preserving Integration

Numerical schemes for the NLSE seek to discretely preserve the energy structure:

• Crank–Nicolson Type Methods: Fully implicit, second-order, energy-preserving schemes that exactly conserve a discrete analogue of the NLSE energy (Besse et al., 2018).
• Relaxation and Generalized Relaxation Methods: Linearly implicit schemes generalizing to arbitrary power-law nonlinearities, designed to conserve a modified discrete energy functional for temporal accuracy and structure preservation (Besse et al., 2018).
• Scalar Auxiliary Variable (SAV) and Integrating-Factor Runge-Kutta (IF-RK): High-order, linearly implicit structure-preserving schemes reformulate the NLSE so that the energy (or a quadratic modified variant) is preserved exactly by design, even after spatial discretization (e.g., via Fourier pseudospectral methods) (Jiang et al., 2021).
• Discontinuous Galerkin (DG) Schemes: Energy-based DG methods defined by mesh-independent numerical fluxes achieve either discrete energy conservation or controlled energy dissipation, governed by flux parameters, and optimal $L^2$-convergence rates (Ren et al., 2023).

6. Analytical and Physical Contexts

The nonlinear Schrödinger energy functional has broad physical and mathematical significance:

• Optical and Gross–Pitaevskii Models: For optical systems with saturable nonlinearities or Bose–Einstein condensates described by the Gross–Pitaevskii equation, the specific energy landscape encodes physically measurable quantities such as chemical potential, ground-state densities, and existence thresholds (Lin et al., 2016, Duaibes et al., 2023).
• Information-Theoretic and Relativistic Corrections: The $q$-deformed energies motivated by maximizing entropy or minimizing relative information provide a bridge to non-classical, information-based formulations of quantum dynamics; parameters such as $\eta$ may encode relativistic corrections (Liew et al., 2009).
• Nonlinear Quantum-Classical Transitions: In models with mass-dependent nonlinear energies, the sign-flip of the kinetic component at a universal critical mass marks a breakdown of quantum spreading in favor of collapse, influencing potential experimental signatures in large-mass interference (Geszti, 2024).

In sum, the nonlinear Schrödinger energy is the organizing variational principle of the NLSE in analysis and computation, fundamentally determining solution structure, stability, dynamical regimes, and numerical robustness across a diverse range of nonlinear wave models.