Schrödinger-Boussinesq System Overview

• The Schrödinger-Boussinesq system is a coupled set of nonlinear PDEs modeling interactions between complex short-wave and real long-wave modes in dispersive media.
• It integrates nonlinear Schrödinger dynamics with Boussinesq-type dispersion, leading to the emergence of solitary waves and a rich variational and spectral structure.
• Analytical techniques, including energy estimates and spectral analysis, establish global well-posedness in 2D and orbital stability in 1D, linking the model to classical Zakharov turbulence.

The Schrödinger-Boussinesq system is a class of coupled nonlinear partial differential equations (PDEs) modeling the interaction between complex short-wave modes governed by a nonlinear Schrödinger equation and real long-wave modes governed by a Boussinesq equation. Such systems arise in the context of nonlinear dispersive media, particularly plasma physics (e.g., Langmuir turbulence), where high-frequency and low-frequency waves interact dynamically. The system is of central interest due to its intricate mathematical structure, which combines features from the nonlinear Schrödinger (NLS), Boussinesq, and Zakharov classes, yielding rich coupling among nonlinear, dispersive, and potential effects.

1. Mathematical Model Formulations

1D Schrödinger–Boussinesq System

The canonical one-dimensional Schrödinger–Boussinesq (SB) system for $$(\varepsilon(t,x), n(t,x)) \in \mathbb{C} \times \mathbb{R}$$ is

$\begin{cases} i\,\varepsilon_t + \varepsilon_{xx} = n\,\varepsilon + \gamma\,|\varepsilon|^2\varepsilon, \[1ex] n_{tt} - n_{xx} + \alpha\,n_{xxxx} - \beta\,(n^2)_{xx} = |\varepsilon|^2_{xx}, \end{cases} \qquad (t,x)\in\mathbb{R}^2,$

with real parameters $\alpha,\,\beta,\,\gamma > 0$ (Ma et al., 26 Dec 2025). The first equation is modified cubic NLS with potential $n$ and nonlinearity $\gamma$. The second is a forced Boussinesq equation with higher-order dispersion ($n_{xxxx}$), quadratic nonlinearity, and external forcing $|\varepsilon|^2_{xx}$. The SB system generalizes prototypical Zakharov-type models, incorporating further Boussinesq dispersive corrections.

2D Schrödinger–improved Boussinesq System

In two spatial dimensions on a domain $\Omega\subset\mathbb{R}^2$, the Schrödinger–improved Boussinesq (S–iB) system has the form

$$\begin{cases} i\,\partial_t u + Au = v\,u, \ a\,\partial_{tt} v - A v - A\,\partial_t v = A\,|u|^2, \end{cases}$$

where $A=-\Delta$ with Dirichlet boundary conditions (Ozawa et al., 2022). The term $-A\,\partial_t v$ represents a Boussinesq-type damping or “improvement”, distinguishing the system from the Zakharov limit.

2. Interpretation of Physical and Mathematical Structure

• Short-wave dynamics: The complex field $\varepsilon$ or $u$ describes envelope-like oscillatory waves, with dispersion via $\varepsilon_{xx}$ (or $Au$) and focusing or defocusing self-interaction through $|\varepsilon|^2\varepsilon$.
• Long-wave sector: The real field $n$ or $v$ models low-frequency perturbations (e.g., ion sound waves, mean field effects), with standard and higher-order Boussinesq dispersive mechanisms by $n_{tt}$, $n_{xx}$, $n_{xxxx}$, and nonlinear self-modulation via $(n^2)_{xx}$ or quadratic nonlinearities.
• Coupling: The product $n\,\varepsilon$ (or $v\,u$) models the effect of the long-wave potential on the short-wave, and $|\varepsilon|^2_{xx}$ (or $A\,|u|^2$) acts as a nonlinear forcing of the long-wave by the short-wave energy density.
• Parameters: $\gamma$ controls the strength of cubic NLS nonlinearity; $\alpha$ and $\beta$ parameterize higher-order dispersion and long-wave nonlinearity; $a$ is the Boussinesq mass parameter.

This interplay results in a complex PDE system exhibiting nonlinear energy transfer, dispersive regularization, and the emergence of various coherent structures, including solitary waves, depending on the precise parameter regime and dimension (Ma et al., 26 Dec 2025, Ozawa et al., 2022).

3. Solitary Waves and Reduction to ODEs

Solitary or traveling wave solutions are sought via the ansatz

$$\varepsilon(t,x) = e^{-i\omega t}\,\phi(x),\quad n(t,x) = \psi(x)$$

imposing decay to zero as $|x|\to\infty$. Substituting this into the SB system and setting $n_{tt} = 0$ gives the stationary coupled ODE system: $$\begin{cases} \phi''(x) - (\psi(x) - \omega)\phi(x) - \gamma|\phi(x)|^2\phi(x) = 0, \ -\,\psi''(x) + \alpha\,\psi^{(4)}(x) - \beta\,(\psi(x)^2)'' = |\phi(x)|^2{}''. \end{cases}$$ Upon integrating the second equation twice under boundary decay, one finds

$$\alpha\,\psi'' - \psi - \beta\,\psi^2 + |\phi|^2 = 0.$$

Solitary wave profiles are then determined as critical points of the Lyapunov functional associated with the underlying Hamiltonian structure (Ma et al., 26 Dec 2025).

4. Variational Structure, Conservation Laws, and Spectral Theory

The SB system admits a variational (Hamiltonian) structure. Key functionals include:

• Hamiltonian (Energy):

$$E(\varepsilon, n, \phi) = \int_{\mathbb{R}} \left\{ |\varepsilon_x|^2 + n\,|\varepsilon|^2 + \tfrac{\gamma}{2}\,|\varepsilon|^4 + \tfrac{1}{2}\,n^2 + \tfrac{\alpha}{2}\,|n_x|^2 + \tfrac{\beta}{3}\,n^3 + \tfrac{1}{2}\,|\phi_x|^2 \right\}\,dx.$$

• Mass (Charge) and Momentum: $$Q_2 = \int |\varepsilon|^2\,dx,\qquad Q_1 = \Im \int \varepsilon_x\,\overline{\varepsilon}\,dx - \int \phi'\,n\,dx.$$
• Lyapunov functional for solitary waves (speed $v$, frequency $\omega$):

$$S_{v,\omega}(\varepsilon,n,\phi) = E(\varepsilon,n,\phi) - v Q_1(\varepsilon,n,\phi) - \omega Q_2(\varepsilon,n,\phi).$$

Solitary waves are constrained minimizers of $S_{v,\omega}$. Conserved quantities play a fundamental role in global well-posedness, stability, and long-time dynamics (Ma et al., 26 Dec 2025, Ozawa et al., 2022).

The spectral properties of the linearized operator around solitary waves are key:

• The operator $H_{v,\omega} = S_{v,\omega}''(\Phi_{v,\omega})$ is self-adjoint with
  • exactly one negative (discrete) eigenvalue,
  • a double zero eigenvalue due to translation and gauge symmetries,
  • essential spectrum bounded away from zero on the positive axis.
  • These facts underpin the Lyapunov–Vakhitov–Kolokolov (VK) mechanism for stability.

5. Existence and Stability of Solitary Waves

Orbital Stability in 1D

For $(\alpha, \beta, \gamma) > 0$ and parameters $(v, \omega)$ and explicit solitary wave profiles (notably those with $\operatorname{sech}$-profiles), the following holds: For all $\varepsilon > 0$, there is $\delta > 0$ such that if initial data are $\delta$-close to a solitary wave profile $\Phi_{v, \omega}$ (in the normed phase space $$X = H^1_{\mathbb{C}} \times H^1_{\mathbb{R}} \times L^2_{\mathbb{R}}$$), then the evolution remains globally close to the modulated solitary wave orbit (modulo translation and phase rotation) for all time. This is established via

• coercivity of the Hessian on constrained directions,
• energetic Lyapunov control,
• conservation of $E$, $Q_1$, $Q_2$,
• spectral decomposition and index-matching with the VK condition,
• construction of a Lyapunov functional $$t \mapsto S_{v,\omega}(\varepsilon(t),n(t),\phi(t)) - d(v,\omega)$$

The result extends previous work on solitary wave stability for coupled Zakharov/Boussinesq systems to the SB context with cubic NLS nonlinearity and quartic Boussinesq dispersion (Ma et al., 26 Dec 2025).

Unique Global Solutions in Higher Dimension

For the 2D S–iB system, global strong solutions exist uniquely for any finite-energy initial data—no $L^2$-smallness is imposed, unlike the Zakharov case. Conservation laws (energy, charge) and a priori exponential bounds (for first and second-order Sobolev norms) are established by direct energy and compactness arguments, notably using Yosida regularization and Aubin–Lions-type compactness (Ozawa et al., 2022).

6. Limiting Behavior, Connections, and Further Properties

• Zakharov Limit: For the S–iB system, setting the improvement parameter to zero recovers the classical Zakharov system, governing Langmuir turbulence. Under a smallness condition $\|u_0\|_2^2 < 2/(a\,C_0)$ (where $C_0$ is the Gagliardo–Nirenberg best constant), one rigorously proves that global solutions of the S–iB system converge to those of the Zakharov system as the improvement parameter vanishes, with convergence in $H^1$, $L^2$, and related spaces for all times in compacts (Ozawa et al., 2022).
• Role of Improvement: The Boussinesq "improvement" term leads to enhanced dispersion and energy control, bypassing the mass threshold phenomena typical for the Zakharov system and allowing global results without size restrictions on the initial data.
• Analytical Techniques: The analysis relies on monotonicity methods, direct energy estimates, spectral analysis for orbital stability, Lyapunov functionals, index counting (VK conditions), and functional analytic compactness methods for well-posedness.
• Relation to Classical Models: Dropping the improvement term returns to Zakharov dynamics. Boussinesq-type corrections improve well-posedness and regularity results compared to NLS or Zakharov alone. In whole-space settings, dispersive/Fourier analytic tools are effective; on domains, compactness and variational methods are preferred.

7. Research Developments and Outlook

The Schrödinger–Boussinesq system represents a unification and generalization of several canonical dispersive models in nonlinear wave and plasma theory. Recent advances have clarified the existence and nonlinear orbital stability of solitary waves in 1D, including a full spectral and variational characterization (Ma et al., 26 Dec 2025), as well as well-posedness and limiting behavior in 2D with energy-critical functional frameworks (Ozawa et al., 2022). These results provide a rigorous mathematical foundation for further investigations on higher-order interactions, global dynamics, blowup scenarios, and the transition from improved Boussinesq to Zakharov-type turbulence in dispersive media.