Well-posedness of the higher-order nonlinear Schrödinger equation on a finite interval

TL;DR

This work establishes local Hadamard well-posedness for the higher-order nonlinear Schrödinger equation on a finite interval with mixed Dirichlet and Neumann boundary data. By blending the unified transform (Fokas method) for the forced linear problem with contraction-mapping arguments in carefully designed function spaces, the authors derive robust Sobolev and Strichartz estimates for both reduced interval problems and half-line problems. They prove high-regularity well-posedness for $\tfrac12 < s \le 2$ under explicit conditions on the nonlinearity exponent $\lambda$, and low-regularity well-posedness for $0 \le s < \tfrac12$ with a restricted $\lambda$ range, all with precise lifespan control and local Lipschitz continuity of the data-to-solution map. A key novelty is the new time-regularity results on the half-line and the intricate linear-decomposition strategy on a finite interval, which together enable a rigorous treatment of a multi-term linear part and a power-type nonlinearity on a bounded domain. These results lay a rigorous foundation for nonlinear dispersive IBVPs on finite intervals and have potential implications for control and applications involving boundary-driven dispersive dynamics.

Abstract

We establish the local Hadamard well-posedness of a certain third-order nonlinear Schrödinger equation with a multi-term linear part and a general power nonlinearity known as the higher-order nonlinear Schrödinger equation, formulated on a finite interval with a combination of nonzero Dirichlet and Neumann boundary conditions. Specifically, for initial and boundary data in suitable Sobolev spaces that are related to one another through the time regularity induced by the equation, we prove the existence of a unique solution as well as the continuous dependence of that solution on the data. The precise choice of solution space depends on the value of the Sobolev exponent and is dictated both by the linear estimates associated with the forced linear counterpart of the nonlinear initial-boundary value problem and, in the low-regularity setting below the Sobolev algebra property threshold, by certain nonlinear estimates that control the Sobolev norm of the power nonlinearity. In particular, as usual in Schrödinger-type equations, in the case of low regularity it is necessary to derive Strichartz estimates in suitable Lebesgue/Bessel potential spaces. The proof of well-posedness is based on a contraction mapping argument combined with the linear estimates, which are established by employing the explicit solution formula for the forced linear problem derived via the unified transform of Fokas. Due to the nature of the finite interval problem, this formula involves contour integrals in the complex Fourier plane with corresponding integrands that contain differences of exponentials in their denominators, thus requiring delicate handling through appropriate contour deformations. It is worth noting that, in addition to the various linear and nonlinear results obtained for the finite interval problem, novel time regularity results are established here also for the relevant half-line problem.

Figures (2)

• Figure 2.1: The open set $\widetilde{D}$ is defined by \ref{['dtil-def']} through the curve $\mathop{\mathrm{Im}}\nolimits(\omega) = 0$ (which, in addition to the real axis, takes the form of a pair of intersecting lines in the case of $\alpha^2 + 3 \beta \delta = 0$ and a hyperbola otherwise), as well as the circle of radius $R_\Delta$ (given by \ref{['rd-def']}) centered at $\frac{\alpha}{3 \beta}$. The positively oriented boundary of $\widetilde{D}$ consists of the nine distinct curves $\Gamma_m$, $m = 1, 2, \ldots, 9$ (labeled only in the second figure for clarity). Additionally, $\widetilde{D}$ is comprised of three connected subsets, $\widetilde{D}_0$, $\widetilde{D}_\pm$, where the subscript refers to the symmetry of $\omega$ (out of $\nu_0=k$ and $\nu_\pm$ given by \ref{['nupm']}) that has positive imaginary part within that subset. Also, in all three cases, the angle $\phi_0$ is equal to half the measure of the arc $\Gamma_2$. Finally, when they exist, the branch points of the square root in the expression \ref{['nupm']} for $\nu_\pm$ are associated with a branch cut taken so that it is entirely contained within the circle $\left\lvert k-\frac{\alpha}{3\beta} \right\rvert = R_\Delta$ and hence lies outside $\widetilde{D}$.
• Figure A.1: The regions $D_0$, $D_+$, $D_-$ depending on the sign of the quantity $\alpha^2 + 3 \beta \delta$.

Theorems & Definitions (41)

• Theorem 1.1: High-regularity Hadamard well-posedness
• Theorem 1.2: Low-regularity Hadamard well-posedness
• Theorem 1.3: Linear estimates
• Remark 2.1
• Theorem 2.1: Sobolev estimate
• Remark 2.2
• proof
• Lemma 2.1: Hardy h1929
• Lemma 2.2
• proof : Proof of Lemma \ref{['lemma:rtotau']}