Inhomogeneous Nonlinear Schrödinger Equation

Updated 30 November 2025

INLSE is a generalized nonlinear Schrödinger equation with spatial or temporal variable coefficients in the nonlinear term and potential, modeling diverse physical contexts.
It exhibits modified scaling laws, critical thresholds for blow-up versus scattering, and intricate dependence on inhomogeneity and nonlinearity exponents.
Advanced techniques such as Strichartz estimates, weighted inequalities, and similarity transformations underpin its rigorous analysis and facilitate the construction of exact solutions.

The inhomogeneous nonlinear Schrödinger equation (INLSE) is a spatially or temporally modulated generalization of the classical NLS, characterized by the presence of variable coefficients in the nonlinear term and/or the potential. The INLSE arises in a wide variety of physical contexts, including nonlinear optics in graded-index fibers, Bose–Einstein condensates with position-dependent interactions, plasma wave modulation, and surface gravity wave propagation over uneven topography. The archetypal focusing INLSE, in $N>2$ ,

$i\,u_t + \Delta u + |x|^{-b} |u|^{p-1} u = 0$

exhibits modified scaling and symmetry properties, sharp thresholds for blow-up/scattering, and intricate dependence on the inhomogeneity exponent $b$ and the nonlinearity exponent $p$ . Advanced techniques including Strichartz estimates, weighted Gagliardo–Nirenberg inequalities, virial methods, concentration–compactness, and supersymmetric quantum mechanics provide a rigorous basis for well-posedness, stability, exact solutions, and dynamical scenarios. Below is a comprehensive survey of the mathematical theory and applications of the INLSE, as established in recent arXiv literature.

1. Canonical Forms and Critical Exponents

The INLSE manifests in several canonical forms, depending on the physical context and the nature of inhomogeneity. The most prominent structural cases are:

Power-law spatial inhomogeneity:

$i\,u_t + \Delta u + |x|^{-b} |u|^{p-1} u = 0 \qquad (x \in \mathbb{R}^N)$

with $b\geq0$ , $p>1$ . The scaling-critical Sobolev index is

$s_c = \frac N2 - \frac{2-b}{p-1}$

defining mass-critical, $L^2$ -critical ( $s_c=0$ ), intercritical ( $0<s_c<1$ ), and energy-critical ( $s_c=1$ ) regimes (Campos, 2019).

General variable coefficient structure:

$i\,u_t + u_{xx} + g(x)\,|u|^2 u = 0$

with $g(x)$ possibly changing sign, realizing sharply localized interfaces between focusing and defocusing nonlinearity (Marangell et al., 2010).

Vector, non-autonomous, and higher-order generalizations, including time or space-time modulation in the nonlinearity, external potential, or gain/loss:

$i\,\psi_t = -\psi_{xx} + V(x) \psi + g(x) |\psi|^2 \psi$

or, for quadratic-cubic interactions:

$i\,\psi_t = -\tfrac12\,\psi_{xx} + V(x,t) \psi + g_2(t)|\psi|\psi + g_3(t)|\psi|^2\psi$

(C. et al., 22 Nov 2025, Cardoso et al., 2017, Ghosh et al., 2021).

The critical exponents for global dynamics and blow-up are determined by the interplay of $b$ , $N$ , and $p$ via scaling. Key invariants are the $L^2$ (mass) and $H^1$ (energy) critical exponents:

$p_{\rm mass\text{-}critical} = 1+\frac{4-2b}{N}, \qquad p_{\rm energy\text{-}critical} = 1+\frac{4-2b}{N-2}$

The intercritical regime $1+\frac{4-2b}{N} < p < 1+\frac{4-2b}{N-2}$ supports a sharp dichotomy between scattering and blow-up, as in the homogeneous case (Campos, 2019).

2. Well-posedness, Scattering, and Blow-Up Theory

Well-posedness and long-time behavior of the INLSE depend sensitively on the spatial inhomogeneity and the regime of nonlinearity. The principal results in the focusing ( $+$ ) case can be summarized as follows:

Global Well-posedness and Scattering Below the Threshold: For initial data $u_0 \in H^1_{rad}(\mathbb{R}^N)$ with

$M[u_0]^{1-s_c} E[u_0]^{s_c} < M[Q]^{1-s_c} E[Q]^{s_c}$

and the corresponding gradient-mass bound, the solution to the focusing INLSE is global and scatters in $H^1$ ; i.e., asymptotically linear behavior as $t \to \pm\infty$ (Campos, 2019). Here, $Q$ is the unique positive radial solution of the stationary problem

$\Delta Q - Q + |x|^{-b} |Q|^{p-1} Q = 0.$

Blow-up and Instability: In critical and supercritical regimes, solutions can blow up in finite time for sufficiently large data or negative energy. For $L^2$ -critical inhomogeneous NLS,

$\|u_0\|_{L^2} < \|Q\|_{L^2} \implies u(t) \text{ global and bounded},$

but equality gives sharp finite-time blow-up, realized by pseudoconformal transformations (Genoud, 2011). In radial or finite-variance initial data, sharp virial identities yield finite-time divergence of the $H^1$ norm (An et al., 2021).

Critical and Supercritical Cauchy Theory: Strichartz estimates, fractional Hardy inequalities, and local smoothing arguments yield local and global well-posedness results. In the energy-critical case,

$i u_t + \Delta u + |x|^{-b} |u|^\sigma u, \quad \sigma = \frac{4-2b}{n-2}$

admits local well-posedness and small data global scattering for $u_0 \in H^1(\mathbb{R}^n)$ , $n > 3$ , $0 < b < 2$ (Lee et al., 2019, An et al., 2021, An et al., 2021).

Supercritical Regimes: In the supercritical intercritical regime, sharp Gagliardo–Nirenberg inequalities provide sufficient (mass-energy) thresholds for global existence and blow-up (Farah, 2016).
Defocusing Case: In the presence of "defocusing" sign, global well-posedness is assured in all critical and subcritical regimes, with scattering shown via virial/Morawetz techniques and weighted Lorentz space bounds (Aloui et al., 2021).

3. Analytical Framework: Estimates, Criteria, and Proof Techniques

The paper of the INLSE utilizes a refined toolkit of functional and harmonic analysis adapted to the inhomogeneous structure:

Strichartz Estimates: Key to contraction-mapping approaches, local well-posedness, and scattering are the inhomogeneous/admissible Strichartz estimates, including weighted forms for handling singular spatial weights $|x|^{-b}$ (Campos, 2019, Lee et al., 2019).
Weighted Gagliardo–Nirenberg Inequalities: The fundamental interpolation inequality for inhomogeneous weights,

$\int |x|^{-b} |u|^{p+1} dx \lesssim \|\nabla u\|_{L^2}^{\theta(p+1)} \|u\|_{L^2}^{(1-\theta)(p+1)},$

underpins threshold estimates and is saturated by the ground state $Q$ (Campos, 2019, Farah, 2016).

Radial Lemmas and Hardy Inequalities: Decay in far-field and estimates near singularities are handled via the Strauss lemma for radial functions and Hardy-type bounds (Campos, 2019).
Virial and Morawetz Identities: Localized virial/Morawetz identities, with weights adapted to inhomogeneity, provide coercivity below the ground state threshold and enable rigorous proof of scattering and mass evacuation (Campos, 2019).
Pseudoconformal Symmetry: The critical $L^2$ -invariant INLSE exhibits pseudoconformal invariance, yielding explicit blow-up solutions and sharp instability of standing waves (Genoud, 2011).
Concentration–Compactness and Rigidity Arguments: Extensions to non-radial settings and endpoint cases are open problems but are expected to require concentration–compactness and profile decomposition technology (Campos, 2019, Lee et al., 2019).

4. Exact Solutions and Spectral Design

Explicit analytic solutions of the INLSE can be constructed via several structural algorithms:

Supersymmetric Quantum Mechanics (SUSY QM): A general procedure to build stationary exact solutions involves mapping the INLSE's stationary reduction

$-\phi''(x) + V(x)\phi(x) + g(x)\phi^3(x) = \mu\phi(x)$

to a standard NLSE using Lie point symmetries and SUSY QM. The superpotential $W(x)$ and its partner potentials generate solvable profiles; for instance, the Pöschl–Teller potential yields

$g(x) = g_0\,\cosh^6(k_0 x), \qquad \phi(x) = (\text{explicit sech-sech composite})$

(C. et al., 22 Nov 2025).

Similarity Transformations: For time- or space-time-modulated coefficients, a similarity (“lens”) transform can reduce the variable-coefficient INLSE to autonomous NLS, enabling construction of solitons, breathers, and rogue waves with amplitude/phase modulation reflecting the inhomogeneity parameters (Cardoso et al., 2017, Manikandan et al., 2014).
Hierarchical and Reversible Transformations: Integrability-preserving gauge/coordinate transformations map forced (non-isospectral) INLSE hierarchies to standard isospectral NLS equations, preserving constraints between dispersion and nonlinear coefficients. This allows for the explicit recovery of accelerating bright and dark solitons in forced systems (Nandy et al., 2019).

5. Inhomogeneity Effects: Physical and Mathematical Consequences

The introduction of spatial or temporal inhomogeneity fundamentally alters the qualitative and quantitative dynamics:

Focusing/Defocusing Regimes and Soliton Nonexistence: Alternating inhomogeneity (e.g., $g(x)$ switching sign or $Q(x)$ changing sign) can preclude globally coherent soliton solutions; focusing/defocusing transitions lead to spatial intervals where classical bright or dark solitons cannot persist due to local sign changes in the effective nonlinear coefficient (Karjanto et al., 2017, Marangell et al., 2010).
Thresholds and Bifurcations: The presence of inhomogeneity creates new critical norms, bifurcation points, and stability conditions (e.g., Vakhitov–Kolokolov slope conditions generalized to inhomogeneous settings). Instability and symmetry-breaking bifurcations are naturally organized by geometric or topological arguments (Marangell et al., 2010).
Stabilization and Modulation: Spatial and temporal modulation (e.g., periodic or localized variations in $g(x)$ or $V(x,t)$ ) can stabilize otherwise unstable localized modes, or generate new classes of robust breathing states under judicious parameter choices (Cardoso et al., 2017).
Interface and Surface Phenomena: In slab or layered inhomogeneity, interface-bound states, asymmetric bifurcations, and composite localized structures are observed, often governed by matching conditions and phase plane geometry (Marangell et al., 2010).
Non-autonomous Control: Non-autonomous INLSEs model the evolution of nonlinear waves under engineered temporal or spatial control, guiding plasmonic energy, matter wave packets, or optical pulses in media with designed refractive index or interaction profiles (Manikandan et al., 2014, Ghosh et al., 2021).

6. Open Problems and Future Directions

Significant avenues for further research and current open problems, as delineated in the literature, include:

Non-radial and Non-symmetric Data: Extension of sharp scattering results to the nonradial setting, possibly via concentration–compactness and rigidity techniques, remains open (Campos, 2019, Lee et al., 2019).
Endpoint and Low-dimensional Cases: Theory for endpoint nonlinearities ( $p$ at the mass- or energy-critical value) and in $N=1,2$ dimensions is incomplete (Aloui et al., 2021).
Critical Regularity and Multi-scale Inhomogeneity: Extending blow-up and global existence criteria to $H^s$ -critical cases with $0An et al., 2021, An et al., 2021).
Higher-order and Vector INLSEs: The development of rigorous theory for higher-order (e.g., biharmonic) INLSEs, vector/multicomponent equations, and systems with balanced gain/loss or non-Hermitian potentials, is an active area (Bai et al., 2023, Ghosh et al., 2021).
Further Spectral Design and Exactly Solvable Models: Expanding the repertoire of exactly solvable inhomogeneous models via SUSY, Lie symmetries, or similarity transformations, notably for complex and PT-symmetric potentials (C. et al., 22 Nov 2025, Ghosh et al., 2021).

7. Representative Results: Theoretical Summary Table

Regime	Main Result	Key Tool(s)
Intercritical, radial	Scattering below sharp threshold	Weighted Strichartz, Virial
$L^2$ -critical	Global vs. blow-up threshold at $\\|Q\\|_{2}$	Gagliardo–Nirenberg, Pseudoconformal
Step/inverted $g(x)$	Bifurcation, symmetry breaking, instability	Phase-plane topology, VK Slope
Non-autonomous coefficients	Stabilization/destabilization of solitons	Similarity transform, numerics
Hierarchy/forced terms	Accelerating soliton solutions	Gauge/coordinate transformation
SUSY QM/shape invariance	Exact localized stationary states	Lie symmetry, SUSY partner
Fourth-order INLSE	Non-radial finite/infinite-time blow-up	Localized virial, G-N Exterior