Interface problems of mixed spatial order

Technical Summary: Interface Mixed Order Problems via the Unified Transform Method

Problem Statement
This work addresses the initial boundary value problem (IBVP) for linear evolution partial differential equations (PDEs) defined on a spatial domain divided into two half-lines, $(-\infty, 0)$ and $(0, \infty)$, separated by an interface at $x=0$. Unlike previous studies where the governing equations on either side of the interface were identical or differed only by scalar coefficients, this paper investigates "mixed order" interface problems. Specifically, it considers the interaction between linear evolution equations of fundamentally different spatial orders and physical behaviors (dispersive vs. dissipative).

The prototypical examples analyzed involve:

1. Left Half-Line ($x < 0$): The linear Schrödinger equation (lS), the heat equation (h), or the biharmonic Schrödinger equation (biS).
2. Right Half-Line ($x > 0$): The Airy equation or the linearized Korteweg de Vries (lKdV) equation.

These systems are coupled via interface conditions at $x=0$. The paper considers two sets of conditions:

• Condition (IfC.1): Continuity of the solution and its first spatial derivative ($C^1$ continuity).
• Condition (IfC.2): Continuity of the solution and its first two spatial derivatives ($C^2$ continuity), required for the fourth-order biharmonic Schrödinger equation.

The interface values (boundary data at $x=0$) are not prescribed a priori; they are unknown functions determined by the coupling of the two distinct PDEs.

Methodology
The authors employ the Unified Transform Method (UTM), also known as the Fokas method, which serves as a generalization of the Fourier transform for IBVPs on domains with boundaries. The solution procedure involves the following steps:

1. Global Relations: For each half-line problem, the authors derive "global relations" (spectral identities) linking the initial data, the solution, and the unknown boundary values in the complex spectral plane.
2. Change of Variables: To handle the mixed orders, the authors introduce specific root-type functions ($\sigma, \rho, \tau, \gamma$) and a biholomorphism $\nu$ (for the lKdV case). These functions map the spectral parameter $\lambda$ to the complex $k$-plane in a way that aligns the dispersion relations of the different equations (e.g., relating $k^2$, $k^3$, and $k^4$).
3. Linear System Construction: By applying these changes of variables to the global relations, the authors construct a linear system (typically $3\times3$ or $4\times4$) for the transforms of the unknown interface values.
4. Solution Representation: Solving this linear system allows the elimination of the unknown interface values. The authors then substitute these expressions back into the integral representations of the solutions for each half-line.
5. Contour Deformation and Asymptotics: A critical technical contribution is the rigorous justification for deforming integration contours (e.g., from the real axis to contours like $\tilde{R}$ in the lower half-plane) to bypass singularities (specifically at $\lambda=0$) and to eliminate integrals involving the time-dependent solution terms ($u(x,t)$, $v(x,t)$). This is achieved through bespoke arguments utilizing the specific asymptotic behavior of the root functions and the exponential kernels, rather than standard Jordan's lemma, due to the mismatch in leading-order terms.

Key Contributions and Results
The paper provides explicit solution formulae for four distinct interface problems, presented as Propositions 1 through 4:

• Proposition 1: Solves the interface between the linear Schrödinger equation and the Airy equation under $C^1$ interface conditions.
• Proposition 2: Solves the interface between the linear Schrödinger equation and the linearized KdV equation (including a lower-order term $av_x$) under $C^1$ conditions. This requires the introduction of the biholomorphism $\nu$ to handle the non-homogeneous dispersion relation.
• Proposition 3: Solves the interface between the heat equation (dissipative) and the Airy equation (dispersive) under $C^1$ conditions.
• Proposition 4: Solves the interface between the biharmonic Schrödinger equation (fourth-order dispersive) and the Airy equation under $C^2$ conditions. This case involves a $4\times4$ linear system due to the higher order of the PDE.

The explicit formulae are expressed as contour integrals involving the initial data transforms and specific kernel functions derived from the root-type mappings.

Numerical Evaluation
To demonstrate the utility of the derived formulae, the authors perform numerical evaluations using Filon quadrature. This method is chosen to handle the highly oscillatory nature of the integrands. The paper presents plots of the solutions for various initial data configurations:

• Schrödinger-Airy: Shows the transition of dispersive waves from the unidirectional Airy regime to the bidirectional Schrödinger regime, and vice versa, highlighting the "roulette-like" patterns and drag effects caused by the interface.
• Biharmonic Schrödinger-Airy: Illustrates similar wave interactions but with dynamics governed by a quartic dispersion relation, leading to faster emergence of complex patterns.
• Heat-Airy: Demonstrates the interaction between diffusive and dispersive waves, showing exponential decay on the heat side and dispersion on the Airy side.

Significance
The paper claims significance in extending the Unified Transform Method to a new class of problems where the governing equations on either side of an interface are of fundamentally different types (different spatial orders and physical natures). Previous applications of the UTM to interface problems were largely restricted to cases where the equations were identical or differed only by scalar coefficients.

By successfully solving these mixed-order problems, the authors demonstrate the robustness of the UTM in handling complex spectral structures arising from disparate dispersion relations. The work provides the first analytical solution framework for such interactions, which are relevant in modeling physical phenomena involving heterogeneous media, such as tumor growth in brain tissue (as cited in the introduction regarding gliomas) or composite materials with varying physical properties. The explicit formulae derived serve as a foundation for future analysis, including potential extensions to nonlinear problems via contraction mapping techniques.