Dynamics of solutions in the 1d bi-harmonic nonlinear Schrödinger equation

1. Problem Statement

The paper investigates the one-dimensional (1D) fourth-order (bi-harmonic) nonlinear Schrödinger (NLS) equation with mixed dispersion:
$$i u_t - \Delta^2 u - 2a \Delta u + |u|^\alpha u = 0, \quad x \in \mathbb{R}, \, t \in \mathbb{R}$$
where $u$ is a complex-valued function, $a \in \mathbb{R}$ is the coefficient of the lower-order (second-order) dispersion, and $\alpha > 0$ is the nonlinearity power.

Key Context:

• Physical Relevance: This equation models intense laser beam propagation in media where higher-order dispersion (quartic) is significant, such as silicon photonic crystal waveguides.
• Classification: The equation is classified as $L^2$-subcritical ($\alpha < 8$), critical ($\alpha = 8$), or supercritical ($\alpha > 8$) based on the scaling index $s = d/2 - 4/\alpha$ (with $d=1$).
• Core Question: The authors aim to understand the global dynamics of solutions, specifically the stability of ground states (solitary waves), the existence of finite-time blow-up, and the asymptotic behavior (scattering vs. blow-up) in the presence of mixed dispersion ($a \neq 0$).

2. Methodology

The study relies heavily on numerical simulations supported by analytical identities and theoretical frameworks.

• Ground State Construction:
  • Exact Solutions: For specific parameter sets ($a < 0$ and specific $\alpha$), explicit solutions involving $\text{sech}$ functions are used for validation.
  • Numerical Construction: For general parameters, stationary solutions $Q(x)$ satisfying $\Delta^2 Q + 2a \Delta Q + b Q - |Q|^\alpha Q = 0$ are computed using a Fourier spectral method combined with Newton-Krylov iteration (GMRES). The domain is treated as a large torus to approximate $\mathbb{R}$.
• Time Evolution:
  • The time-dependent equation is solved using Exponential Time Differencing (ETD) schemes, specifically the Cox-Matthews scheme (4th order), which is efficient for stiff systems arising from the fourth-order derivative.
  • Conservation Laws: The numerical accuracy is monitored via the conservation of Mass ($M[u]$) and Energy ($E[u]$).
• Analysis of Dynamics:
  • Perturbation Analysis: Ground states are perturbed by scaling factors $A \approx 1$ ($u_0 = A Q$) to test stability.
  • Bifurcation Analysis: The relationship between Energy ($E$) and Mass ($M$) is plotted to identify branching phenomena (stable vs. unstable branches).
  • Blow-up Analysis: Near the blow-up time $t^*$, the authors employ dynamic rescaling to analyze the blow-up rate and profile, comparing numerical results against self-similar ansatzes.

3. Key Contributions and Results

A. Ground State Bifurcation and Stability (Subcritical & Critical Cases)

A major discovery is the existence of two branches of ground states for certain ranges of nonlinearity power $\alpha$ (specifically $2 \le \alpha^* < \alpha < \alpha^* < 10$) when mixed dispersion is present ($a \neq 0$).

• Branching Phenomenon: For a fixed mass, there exist two distinct ground state solutions:
  1. Unstable Branch: Higher energy. Perturbations with slightly increased mass ($A > 1$) "jump" to the stable branch. Perturbations with decreased mass ($A < 1$) disperse (scatter to zero).
  2. Stable Branch: Lower energy. Perturbations asymptotically approach a rescaled version of this ground state.
• Significance: This breaks the standard dichotomy (scattering vs. blow-up) seen in the classical NLS. Instead, a trichotomy emerges:
  1. Scattering to zero (dispersion).
  2. Asymptotic approach to a stable soliton (scattering to a ground state).
  3. Finite-time blow-up.

B. Dynamics in the Critical Case ($\alpha = 8$)

The critical case exhibits the most complex behavior due to the broken scaling invariance when $a \neq 0$.

• The "Gap" Phenomenon: Unlike the standard NLS where mass $M[u_0] > M[Q]$ implies blow-up, the mixed dispersion case shows a gap.
  • If $M[u_0]$ is slightly above the mass of the unstable ground state $Q(a)$, the solution does not immediately blow up. Instead, it may jump to a stable branch or approach a different final state.
  • Blow-up only occurs when the mass exceeds a higher threshold (often related to the mass of the scaling-invariant ground state $Q(0)$ where $a=0$).
• Self-Similar Blow-up: When blow-up occurs, it follows a self-similar profile.
  • Profile: The blow-up profile converges to the ground state of the pure quartic case ($a=0$), denoted $Q(0)$, regardless of the value of $a$.
  • Rate: The blow-up rate for the $L^\infty$ norm is found to be $\|u(t)\|_{L^\infty} \sim (t^* - t)^{-1/6}$, and for the $H^2$ norm (related to $\|\Delta u\|_{L^2}$), it is $\sim (t^* - t)^{-2/3}$. This differs from conjectured rates in previous literature for the pure case.

C. Dynamics in the Supercritical Case ($\alpha > 8$)

• Dichotomy Restored: In the supercritical regime (e.g., $\alpha = 10$), the branching phenomenon disappears. The system reverts to a standard dichotomy:
  • $M[u_0] < M[Q] \implies$ Scattering (dispersion).
  • $M[u_0] > M[Q] \implies$ Finite-time blow-up.
• Blow-up Profile: The blow-up profile is distinct from the ground state $Q(0)$ and converges to a localized smooth solution of the profile equation. The blow-up rate is faster, following $\|u(t)\|_{L^\infty} \sim (t^* - t)^{-1/\alpha}$.

D. Oscillatory Nature of Ground States

• For $a > 0$ (positive lower dispersion), ground states are non-monotonic and oscillatory, changing sign as $|x| \to \infty$.
• For $a \le -\sqrt{b}$, ground states are positive and monotonic (radially decreasing).
• The transition between these regimes is linked to the stability properties and the bifurcation of branches.

4. Significance and Conclusions

• Novelty of Mixed Dispersion: The paper demonstrates that adding a lower-order dispersion term ($a \neq 0$) fundamentally alters the dynamics of the bi-harmonic NLS, creating a "gap" in the scattering/blow-up threshold and introducing stable/unstable branches of solitons.
• Soliton Resolution: The results support the soliton resolution conjecture for generic initial data in the subcritical case, where solutions decompose into a stable soliton and radiation.
• Blow-up Universality: A striking finding is that the blow-up profile in the critical case is universal, determined solely by the scaling-invariant ($a=0$) ground state, even when the equation contains mixed dispersion.
• Experimental Relevance: These findings provide a mathematical framework for interpreting recent experiments on pure quartic solitons in photonic crystals, suggesting that stability and blow-up behaviors are highly sensitive to the interplay between second-order and fourth-order dispersion.

In summary, the paper establishes that the 1D bi-harmonic NLS with mixed dispersion exhibits a rich dynamical landscape characterized by ground state bifurcation, a trichotomy of long-term behaviors, and a unique universality in blow-up profiles, challenging the standard dichotomy observed in classical NLS equations.