The sine Gordon equation in light-cone coordinates on the half lines revisited: a Riemann--Hilbert approach

Technical Summary: The Sine-Gordon Equation in Light-Cone Coordinates on the Half Lines Revisited

Problem Statement
This work investigates the Initial Boundary Value (IBV) problems for the sine-Gordon (sG) equation in light-cone coordinates, $u_{xt} = \sin u$, posed on two distinct half-line domains: the "right problem" ($x \ge 0, t \ge 0$) and the "left problem" ($x \le 0, t \ge 0$). The study assumes suitable decay conditions as $|x| \to \infty$ and utilizes specific functional spaces, including $H^{2,2}$ for initial data and $\tilde{H}^{1,2}$ for the solution regularity. The primary objective is to determine the well-posedness of these problems, specifically identifying the minimal set of boundary and initial data required to uniquely determine the solution in each domain.

Methodology
The analysis employs the Riemann–Hilbert (RH) problem framework, building upon the Unified Transform Method (UTM) introduced by Fokas. The methodology proceeds through the following stages:

1. Lax Pair and Eigenfunctions: The sG equation is treated as the compatibility condition of a Lax pair. The author constructs Jost solutions (eigenfunctions) $\Phi_j$ and $\Phi_{0j}$ associated with the Lax pair near spectral parameters $k = \infty$ and $k = 0$. These eigenfunctions are defined via Volterra-type integral equations with integration contours adapted to the specific half-line geometry (e.g., integrating from $+\infty$ for the right problem and $-\infty$ for the left problem).
2. Spectral Functions: Direct spectral mappings are established to relate the initial data $u(x,0)$ and boundary data $u(0,t)$ to spectral functions ($a(k), b(k), A(k), B(k)$ and their tilded counterparts). These functions satisfy determinant relations and symmetry properties derived from the traceless nature of the Lax coefficients.
3. Global Relations and Compatibility: The paper derives "Global Relations" (asymptotic relations among spectral functions) that reflect the compatibility between initial and boundary data. For the right problem, these relations are shown to be automatically satisfied by the properties of the spectral functions. For the left problem, they impose specific constraints.
4. Inverse Spectral Mappings: Inverse mappings are formulated as RH problems. The jump matrices for these problems are constructed from the spectral functions. The paper distinguishes between "master" RH problems (parametrized by both $x$ and $t$) and auxiliary RH problems (parametrized by $x$ or $t$ alone).
5. Reconstruction and Regularity: Solutions to the sG equation are recovered from the solution of the master RH problem by analyzing the asymptotic behavior of the RH solution at $k \to \infty$ and $k \to 0$. The differentiability of the RH solution with respect to $x$ and $t$ is rigorously established using $L^2$-RH theory and singular integral equations, ensuring the reconstructed function satisfies the sG equation and the required regularity classes.

Key Contributions and Results

• Distinct Well-Posedness Conditions: The central finding is a significant difference in the well-posedness requirements for the two half-line problems:

  • Right Problem ($x \ge 0$): The solution is uniquely determined by the initial data alone ($u(x,0)$ for $x \ge 0$). The boundary value $u(0,t)$ is not an independent input but is implicitly determined by the evolution of the initial profile. The master RH problem for this case relies solely on the spectral functions associated with the initial data.
  • Left Problem ($x \le 0$): The solution is not uniquely determined by the initial data alone. In addition to $u(x,0)$ for $x \le 0$, the boundary value $u(0,t)$ must be prescribed as an independent input to obtain a well-posed problem. The master RH problem for this case requires spectral functions derived from both the initial and boundary data.

• Existence and Uniqueness Theorems:

  • For the right problem, the paper proves that for initial data $u_0 \in H^{2,2}((0, \infty)) + 2\pi k$, there exists a unique solution $u(x,t) \in C^1([0,T], \tilde{H}^{1,2}((0, \infty)))$ constructed via the RH problem.
  • For the left problem, given initial data $u_0 \in H^{2,2}((-\infty, 0)) + 2\pi k$ and boundary data $v_0 \in H^1(0,T)$ satisfying the compatibility condition $u_0(0) = v_0(0)$, there exists a unique solution $u(x,t) \in C^1([0,T], \tilde{H}^{1,2}((-\infty, 0)))$.

• Asymptotic Behavior: The paper analyzes the behavior of the reconstructed solutions as $x \to +\infty$ and $x \to -\infty$. By extending the initial data to the whole line or utilizing specific decay properties of the reflection coefficients, it is shown that the solutions satisfy the required decay conditions in the respective half-planes.

Significance
The paper addresses open questions regarding the well-posedness of the sine-Gordon equation on half-lines, particularly concerning the necessity of boundary data. While previous literature (e.g., Fokas, Leon, Sakhnovich) has treated IBV problems for sG, this work clarifies a fundamental asymmetry between the two half-line configurations. It demonstrates that the "right problem" behaves more like a Cauchy problem on a half-line where the boundary is determined by the interior dynamics, whereas the "left problem" requires explicit boundary control. This distinction is derived rigorously through the spectral analysis of the associated Lax pair and the construction of specific Riemann–Hilbert factorizations. The results provide a complete characterization of the data required for well-posedness in both light-cone coordinate settings.