Fokas Unified Transform Method

• Fokas Unified Transform Method is an analytical framework for solving initial-boundary value problems for linear and integrable nonlinear PDEs by unifying Fourier, Laplace, and inverse scattering techniques.
• The method constructs explicit solution representations using complex contour integrals or Riemann–Hilbert problems, encoding all initial and boundary data through spectral transforms and global relations.
• It generalizes classical transform methods by accommodating non-self-adjoint operators, arbitrary boundary conditions, and complex domains, thereby advancing theory and computational practices in evolution PDEs.

The Fokas Unified Transform Method ("UTM") is an analytical framework for solving initial-boundary value problems (IBVPs) for linear and integrable nonlinear partial differential equations (PDEs), notable for systematically unifying and extending Fourier, Laplace, and inverse scattering transform (IST) techniques. Central to the method is the construction of explicit representations of the solution in terms of complex contour integrals or, for integrable nonlinear systems, Riemann–Hilbert (RH) problems, with all dependence on the initial and boundary data encoded through spectral transforms and matrix relations known as global relations. The UTM fundamentally generalizes classical transform methods by incorporating non-self-adjoint operators, arbitrary boundary conditions, and domains such as the half-line, intervals, or more complex graphs, and remains applicable to both linear and integrable nonlinear PDEs, including coupled and higher-order systems.

1. Lax Pair Formalism, Spectral Analysis, and the Global Relation

For integrable (nonlinear) PDEs, the UTM is founded on the existence of an associated Lax pair—two compatible linear equations for a spectral function $\psi(x, t, \lambda)$, parametrized by a complex spectral variable $\lambda$. For instance, the coupled Fokas–Lenells system on the half-line $x>0$ employs a $3\times 3$ Lax pair,

$$\psi_x = U(x, t, \lambda)\, \psi, \qquad \psi_t = V(x, t, \lambda)\, \psi,$$

with $U$ and $V$ rational in $\lambda$, depending on both the fields and their $x$-derivatives, and encoding integrability via compatibility $U_t - V_x + [U, V] = 0$ (Hu et al., 2017). The linear case proceeds analogously but with simpler $U,V$ structures.

To solve the IBVP, one introduces matrix-valued eigenfunctions, defined via Volterra (integral) equations used as the basis for spectral analysis. For the coupled Fokas–Lenells system, these eigenfunctions $\mu_j(x, t, \lambda)$, normalized at specific initial or boundary points, possess distinct analyticity properties in regions of the complex $\lambda$-plane, dictated by the decay of exponential kernels in the Volterra construction. The $\lambda$-plane is partitioned into open sectors $D_n$; the analyticity properties underpin the formulation of a Riemann–Hilbert problem. The spectral dependence of eigenfunctions directly reflects the PDE's structure and the domain geometry (Hu et al., 2017, Caudrelier, 2017, Shepelsky et al., 2023, Hu et al., 2017).

Central to the method is the global relation: an algebraic matrix equation coupling spectral transforms of the initial and boundary data, derived from the compatibility and analyticity of the Lax pair eigenfunctions. This global relation, such as $$S^{-1}(\lambda) s(\lambda) = e^{-2ik^2T \hat{\sigma}} \mu_3(0,T,\lambda)$$ (for the Fokas–Lenells case), encodes the mutual dependence of initial and boundary conditions in the spectral domain. For linear constant-coefficient problems, the global relation is a direct algebraic constraint among Fourier/Laplace transforms of the initial and boundary data (Smith, 2014, Kalimeris et al., 20 Oct 2025).

2. Riemann–Hilbert Problem Construction and Analyticity Structure

The solution to the IBVP is constructed as a sectionally analytic (or meromorphic) matrix function $M(x,t,\lambda)$, with boundary values on the sector boundaries of the $\lambda$-plane related by explicit jump matrices. In the Fokas–Lenells system, $M(x,t,\lambda)$ is assembled from the eigenfunctions $\mu_j$ in various sectors, and its jumps across $\overline{D_n} \cap \overline{D_m}$ are determined by the ratio of scattering matrices,

$$M_n = M_m J_{m, n}(x, t, \lambda), \qquad J_{m, n}(x, t, \lambda) = e^{i\lambda^2 x \hat{\sigma} + 2ik^2 t \hat{\sigma}} (S_m^{-1} S_n),$$

where $S_n$ are sector-dependent spectral data. Poles in $M$ arise at zeros of determinants of blocks in the scattering matrices, and explicit residue conditions are imposed at these discrete eigenvalues, reflecting the algebraic spectral structure of the IBVP (Hu et al., 2017, Shepelsky et al., 2023).

For linear problems, such as constant-coefficient evolution equations, the analogue is a single scalar or vector-valued spectral transform, with inversion and contour deformation governed by analyticity in $\lambda$ and rational spectral symmetries (Smith, 2014, Smith, 2022, Kalimeris et al., 20 Oct 2025).

3. Extraction of the Physical Solution and Data Dependence

With the RH problem (or, for linear equations, the contour integral) formulated, the physical fields are recovered from the asymptotic ($\lambda\to\infty$) expansion: $$M(x,t,\lambda) = \mathbb{I} + \lambda^{-1} M^{(1)}(x,t) + \mathcal{O}(\lambda^{-2}),$$ and, for the Fokas–Lenells system,

$$q_x(x,t) = -2i \lim_{\lambda\to\infty} [\lambda M(x,t,\lambda)]_{12}, \qquad r_x(x,t) = -2i \lim_{\lambda\to\infty} [\lambda M(x,t,\lambda)]_{13},$$

with $q(x,t), r(x,t)$ then reconstructed by $x$-integration from prescribed boundary values. For other systems (such as the coupled modified NLS (Hu et al., 2017) or periodic NLS (Shepelsky et al., 2023)), similar expansions yield the solution via the leading off-diagonal terms. For linear systems, the inversion or the large-$\lambda$ limit of the spectral object recovers the solution directly.

The data entering the RH or integral representation are completely determined by the initial and boundary values, modulo the global relation: only those combinations consistent with the analyticity and spectral constraints encoded in the global relation are admissible (Hu et al., 2017, Caudrelier, 2017, Shepelsky et al., 2023, Kalimeris et al., 20 Oct 2025).

4. Symmetry, Reduction, and Generalization

The UTM formalism systematically incorporates the symmetries of the underlying Lax pair and boundary geometry. For example, the involutive reduction $$\psi^{-1}(x,t,\lambda) = A \overline{\psi(x,t,\bar{\lambda})}^T A$$ imposes constraints on the scattering data, essential for correctly defining the Riemann–Hilbert problem (Hu et al., 2017). For star-graph (network) problems, the matrix generalization of UTM leads to $2N \times 2N$ Lax pairs, displaying full algebraic compatibility with domain topology (Caudrelier, 2017).

The method also recovers well-known results as special cases. The full-line (Cauchy problem) IST is retrieved as a symmetric case of the half-line UTM with linearizable boundary conditions—see the embedding of the full-line AKNS problem as a 4$\times$4 half-line problem under appropriate parity and boundary symmetries (Caudrelier, 2017). Reductions, such as the $\mathbb{Z}_2$ reduction group of Mikhailov, allow for the simultaneous treatment of local and nonlocal NLS systems within the UTM framework simply by altering the action of this symmetry on the Lax pair and boundary reflection data.

5. Application to Periodic and Finite-Band Problems

For periodic or quasi-periodic problems—such as periodic finite-band solutions of focusing NLS—the UTM produces RH problems with jump matrices characterized by arc endpoints (branch points) and constant phases. The spectral arcs $\Sigma_j$ correspond to the cuts of a hyperelliptic Riemann surface, and phases $\phi_j$ serve as direct spectral invariants. The solution is reconstructed from the matrix solution $\Psi(z)$ of the RH problem on these arcs,

$$q(x,t) = 2i \lim_{z \to \infty} [z \Psi_{12}(x,t;z)],$$

where the phase recovery reduces to solving a Cauchy-type scalar RH problem, whose normalization provides a linear system for the $\phi_j$ in terms of the solution's periodic sample (Shepelsky et al., 2023).

6. Integration with Linear Spectral Theory and Extensions

For linear problems, the UTM generalizes the eigenfunction expansion method to non-self-adjoint and even nonlocal or nonclassical boundary conditions, using contour integrals instead of series. The UTM's spectral functionals are not ordinary eigenfunctions but "augmented eigenfunctions"—functionals which "diagonalize" the spatial operator up to an explicit boundary term. Upon inversion, the contribution of these boundary functionals vanishes, leading to a representation that genuinely reduces the PDE to an ODE in time for spectral data (Smith, 2014, Smith, 2022).

7. Conclusion and Broader Impact

The Fokas Unified Transform Method structurally unifies the solution of IBVPs for both linear and integrable nonlinear PDEs under a spectral framework that encodes initial and boundary data in a systematic, algorithmic formulation. It bypasses the limitations of classical transform techniques—such as lack of completeness of eigenfunctions or the constraint of self-adjointness—and is applicable to higher-order, coupled, and matrix systems, arbitrary domain geometries (half-line, finite intervals, star graphs), and compatible with general boundary and interface data. The method has also been adapted to semi-discrete and fractional-order systems, as well as to the numerical computation of solutions with uniform accuracy throughout the domain (Cisneros et al., 2021, Fernandez et al., 2018, Deconinck et al., 2020). This provides a universal, analytically rigorous, and computationally effective framework for a vast range of evolution PDEs (Hu et al., 2017, Caudrelier, 2017, Shepelsky et al., 2023, Smith, 2014, Kalimeris et al., 20 Oct 2025).