Zakharov–Shabat System Overview

• Zakharov–Shabat system is a first-order 2×2 spectral problem that underpins the inverse scattering transform for integrable nonlinear PDEs like the NLS and mKdV equations.
• It organizes Jost solutions, scattering data, and monodromy into a unified framework, enabling analysis via semiclassical quantization and differential Galois theory.
• The system extends to generalized AKNS and gauge-equivalent formulations, with applications in numerical spectral computation, integrable probability, and geometric reconstructions.

Searching arXiv for recent and foundational papers on the Zakharov–Shabat system to ground the article in current literature. The Zakharov–Shabat system is a first-order $2\times 2$ spectral problem of the form

$$w_x=\begin{pmatrix}-ik & q(x)\ r(x) & ik\end{pmatrix}w,\qquad w\in\mathbb C^2,$$

with spectral parameter $k\in\mathbb C$ and potentials $q(x),r(x)$. In this form it is the linear scattering problem underlying the inverse scattering transform for several integrable nonlinear PDEs, including nonlinear Schrödinger, modified KdV, sine-Gordon, and sinh-Gordon equations; in broader usage it also includes generalized AKNS systems, gauge-equivalent formulations, and matrix or Lie-algebraic extensions (Yagasaki, 8 Jun 2025, Yagasaki, 2021, Grahovski, 2011). The system organizes Jost solutions, scattering coefficients, discrete spectrum, Stokes data, and monodromy into a single spectral framework, and current work treats it simultaneously as an object of inverse scattering, differential Galois theory, semiclassical analysis, numerical spectral computation, and integrable probability (Yagasaki, 8 Jun 2025, Dong et al., 8 May 2025, Baik et al., 2018).

1. Canonical form, reductions, and associated nonlinear equations

The basic decaying-potential setting assumes

$$\lim_{x\to\pm\infty}q(x)=\lim_{x\to\pm\infty}r(x)=0,$$

with $q,r$ meromorphic near the real axis in one line of work and holomorphic in a neighborhood of $\mathbb R$ with exponential-coordinate expansions in another (Yagasaki, 8 Jun 2025, Yagasaki, 2021). A common family of symmetry reductions is

$r=q,\qquad r=-q,\qquad r=q^\ast,\qquad r=-q^\ast,$

which distinguishes several physically and analytically important regimes (Yagasaki, 8 Jun 2025).

Within the inverse scattering transform, the ZS system serves as the spectral problem for several standard integrable PDEs. The correspondences stated in the literature are as follows.

Nonlinear PDE	Reduction or associated relation
NLS	$iq_t=q_{xx}\pm 2|q|^2q,\quad r=\mp q^\ast$
mKdV	$q_t\pm 6q^2q_x+q_{xxx}=0,\quad r=\mp q$
sine-Gordon	$$w_x=\begin{pmatrix}-ik & q(x)\ r(x) & ik\end{pmatrix}w,\qquad w\in\mathbb C^2,$$0
sinh-Gordon	$$w_x=\begin{pmatrix}-ik & q(x)\ r(x) & ik\end{pmatrix}w,\qquad w\in\mathbb C^2,$$1
KdV-related scalar problem	reducible to a Schrödinger equation

These identifications place the ZS system at the center of the AKNS scattering formalism rather than at the periphery of any one equation (Yagasaki, 2021). In the semiclassical defocusing setting, after removing the fast phase from $$w_x=\begin{pmatrix}-ik & q(x)\ r(x) & ik\end{pmatrix}w,\qquad w\in\mathbb C^2,$$2, one obtains the self-adjoint ZS system

$$w_x=\begin{pmatrix}-ik & q(x)\ r(x) & ik\end{pmatrix}w,\qquad w\in\mathbb C^2,$$3

with

$$w_x=\begin{pmatrix}-ik & q(x)\ r(x) & ik\end{pmatrix}w,\qquad w\in\mathbb C^2,$$4

which is the form relevant to the defocusing NLS inverse-scattering problem (Dong et al., 8 May 2025).

A distinct periodic formulation on the circle uses the non-selfadjoint operator

$$w_x=\begin{pmatrix}-ik & q(x)\ r(x) & ik\end{pmatrix}w,\qquad w\in\mathbb C^2,$$5

with real analytic $$w_x=\begin{pmatrix}-ik & q(x)\ r(x) & ik\end{pmatrix}w,\qquad w\in\mathbb C^2,$$6-periodic $$w_x=\begin{pmatrix}-ik & q(x)\ r(x) & ik\end{pmatrix}w,\qquad w\in\mathbb C^2,$$7; this setting emphasizes monodromy and exact WKB quantization rather than scattering on the line (Fujiié et al., 2017). The same spectral architecture therefore supports both decaying and periodic problems, self-adjoint and non-selfadjoint reductions, and line- versus circle-based spectral theories.

2. Jost solutions, scattering data, and inverse scattering structure

For decaying potentials, the ZS system admits Jost solutions $$w_x=\begin{pmatrix}-ik & q(x)\ r(x) & ik\end{pmatrix}w,\qquad w\in\mathbb C^2,$$8 with standard asymptotics

$$w_x=\begin{pmatrix}-ik & q(x)\ r(x) & ik\end{pmatrix}w,\qquad w\in\mathbb C^2,$$9

$k\in\mathbb C$0

and scattering relations

$k\in\mathbb C$1

The reflection coefficients are

$k\in\mathbb C$2

when $k\in\mathbb C$3 (Yagasaki, 8 Jun 2025).

Because the coefficient matrix has zero trace, the Wronskian is constant, and one obtains

$k\in\mathbb C$4

hence

$k\in\mathbb C$5

This identity is structurally important in both scattering theory and differential Galois theory (Yagasaki, 8 Jun 2025).

The potentials are called reflectionless if

$k\in\mathbb C$6

equivalently $k\in\mathbb C$7 on the continuous spectrum (Yagasaki, 8 Jun 2025, Yagasaki, 2021). In inverse-scattering language, reflectionless potentials are precisely those with no continuous scattering component, so only discrete spectrum contributes; they are the soliton-type potentials (Yagasaki, 8 Jun 2025). A recurring misconception is that reflectionless means trivial. The literature states the opposite: reflectionless data generate the familiar soliton sector, including $k\in\mathbb C$8-soliton solutions (Yagasaki, 2021).

The inverse problem is commonly formulated through Gelfand–Levitan–Marchenko equations. For the left formulation,

$k\in\mathbb C$9

$q(x),r(x)$0

while for the right formulation,

$q(x),r(x)$1

$q(x),r(x)$2

These two formulations are spectrally equivalent but numerically distinct. Their scattering coefficients satisfy

$q(x),r(x)$3

and for the $q(x),r(x)$4-soliton case the norming constants obey

$q(x),r(x)$5

(Chernyavsky et al., 2022). This left/right duality matters computationally because one can transform the right problem into an auxiliary left problem by

$q(x),r(x)$6

thereby reusing left-formulation solvers (Chernyavsky et al., 2022).

3. Solvability by quadrature and the reflectionless criterion

A central recent development is the characterization of when the ZS system is solvable by quadrature in the sense of differential Galois theory. In this setting, “integrable” means that the solution field is Liouvillian, equivalently that the linear system is solvable by quadratures; it does not mean merely that the associated nonlinear PDE belongs to the general soliton-theory class (Yagasaki, 8 Jun 2025, Yagasaki, 2021). That distinction is explicit in the literature and removes a common ambiguity.

Under the assumptions that $q(x),r(x)$7 are meromorphic near $q(x),r(x)$8, decay at $q(x),r(x)$9, and satisfy

$$\lim_{x\to\pm\infty}q(x)=\lim_{x\to\pm\infty}r(x)=0,$$0

the 2025 theorem proves the equivalence

$$\lim_{x\to\pm\infty}q(x)=\lim_{x\to\pm\infty}r(x)=0,$$1

with the converse additionally assuming analyticity near $$\lim_{x\to\pm\infty}q(x)=\lim_{x\to\pm\infty}r(x)=0,$$2 on the Riemann sphere and one of the symmetry reductions above (Yagasaki, 8 Jun 2025). An earlier result established the same equivalence for a wide class of analytic decaying potentials satisfying condition (A), where $$\lim_{x\to\pm\infty}q(x)=\lim_{x\to\pm\infty}r(x)=0,$$3 have exponential-coordinate expansions and are $$\lim_{x\to\pm\infty}q(x)=\lim_{x\to\pm\infty}r(x)=0,$$4 (Yagasaki, 2021).

The differential-Galois argument passes through the transformed system $$\lim_{x\to\pm\infty}q(x)=\lim_{x\to\pm\infty}r(x)=0,$$5,

$$\lim_{x\to\pm\infty}q(x)=\lim_{x\to\pm\infty}r(x)=0,$$6

which has an irregular singularity at $$\lim_{x\to\pm\infty}q(x)=\lim_{x\to\pm\infty}r(x)=0,$$7 when $$\lim_{x\to\pm\infty}q(x)=\lim_{x\to\pm\infty}r(x)=0,$$8 are analytic near $$\lim_{x\to\pm\infty}q(x)=\lim_{x\to\pm\infty}r(x)=0,$$9 (Yagasaki, 8 Jun 2025). The formal exponential part is

$q,r$0

and the Stokes matrices are identified with the scattering data: $q,r$1 Ramis’ theorem then implies that the differential Galois group contains the Zariski closure of the group generated by the formal monodromy, the exponential torus, and the Stokes matrices. Since the formal monodromy is trivial and the exponential torus is diagonal, integrability by quadrature forces strong restrictions on the off-diagonal Stokes entries, leading to

$q,r$2

and finally to

$q,r$3

by analyticity and symmetry (Yagasaki, 8 Jun 2025).

The constructive direction is equally important. In the reflectionless case, the Jost solutions can be reconstructed from the discrete poles of $q,r$4 and $q,r$5 by residue calculus and projection operators after introducing

$q,r$6

As a result, the solutions are expressed using rational operations, exponentials $q,r$7, and differentiation, which is exactly solvability by quadrature (Yagasaki, 8 Jun 2025). In the earlier analytic-decay framework, reflectionless potentials were further shown to be rational functions of $q,r$8 for some $q,r$9 with $\mathbb R$0 (Yagasaki, 2021).

4. Semiclassical quantization, periodic spectra, and pseudospectra

In the self-adjoint semiclassical ZS problem, turning points are defined by

$\mathbb R$1

For

$\mathbb R$2

the combinations

$\mathbb R$3

give a factorization

$\mathbb R$4

When there are exactly two simple real turning points $\mathbb R$5 with $\mathbb R$6 between them, the eigenvalues satisfy the perturbed Bohr–Sommerfeld rule

$\mathbb R$7

and each true eigenvalue lies within $\mathbb R$8 of a quantized level (Dong et al., 8 May 2025). The proof uses a comparison-equation framework for traceless $\mathbb R$9 first-order systems and a Weber model

$r=q,\qquad r=-q,\qquad r=q^\ast,\qquad r=-q^\ast,$0

which is uniform from the bottom of the well up to finite $r=q,\qquad r=-q,\qquad r=q^\ast,\qquad r=-q^\ast,$1 (Dong et al., 8 May 2025).

For the non-selfadjoint ZS operator on the circle,

$r=q,\qquad r=-q,\qquad r=q^\ast,\qquad r=-q^\ast,$2

exact WKB analysis yields two semiclassical quantization regimes. Without real turning points, periodic eigenvalues satisfy

$r=q,\qquad r=-q,\qquad r=q^\ast,\qquad r=-q^\ast,$3

hence

$r=q,\qquad r=-q,\qquad r=q^\ast,\qquad r=-q^\ast,$4

With real turning points $r=q,\qquad r=-q,\qquad r=q^\ast,\qquad r=-q^\ast,$5, the trace of monodromy leads to

$r=q,\qquad r=-q,\qquad r=q^\ast,\qquad r=-q^\ast,$6

and therefore

$r=q,\qquad r=-q,\qquad r=q^\ast,\qquad r=-q^\ast,$7

for some $r=q,\qquad r=-q,\qquad r=q^\ast,\qquad r=-q^\ast,$8 (Fujiié et al., 2017).

On the line, the non-selfadjoint semiclassical ZS operator exhibits a large pseudospectrum. After the gauge change associated with $r=q,\qquad r=-q,\qquad r=q^\ast,\qquad r=-q^\ast,$9, the principal symbol is

$iq_t=q_{xx}\pm 2|q|^2q,\quad r=\mp q^\ast$0

and the pseudospectral region is characterized geometrically by

$iq_t=q_{xx}\pm 2|q|^2q,\quad r=\mp q^\ast$1

Inside this region the system admits compactly supported quasimodes with residual smaller than any power of $iq_t=q_{xx}\pm 2|q|^2q,\quad r=\mp q^\ast$2, whereas outside it the resolvent is uniformly bounded (VanValkenburgh, 2013). This pseudospectral picture is directly relevant to the numerical computation of true eigenvalues because small perturbations can create eigenvalues where the resolvent is large (VanValkenburgh, 2013).

5. Numerical analysis and computational methods

A large numerical literature treats the direct ZS problem,

$iq_t=q_{xx}\pm 2|q|^2q,\quad r=\mp q^\ast$3

as the computational core of the nonlinear Fourier transform for the nonlinear Schrödinger equation (Medvedev et al., 2019, Medvedev et al., 2019). For real $iq_t=q_{xx}\pm 2|q|^2q,\quad r=\mp q^\ast$4, the quadratic invariant

$iq_t=q_{xx}\pm 2|q|^2q,\quad r=\mp q^\ast$5

is conserved, and conservative schemes are designed to preserve this property exactly or discretely (Medvedev et al., 2019, Medvedev et al., 2019, Medvedev et al., 2020).

Two fourth-order exponential one-step schemes, ES4 and TES4, were constructed from a Magnus-type expansion. ES4 has

$iq_t=q_{xx}\pm 2|q|^2q,\quad r=\mp q^\ast$6

while TES4 factors the step as

$iq_t=q_{xx}\pm 2|q|^2q,\quad r=\mp q^\ast$7

A distinctive feature of TES4 is that the spectral parameter appears only in the middle exponential, which allows fast computational algorithms (Medvedev et al., 2019). A separate fourth-order conservative scheme CT4 generalizes the Boffetta–Osborne method by transforming out local dynamics and using neighboring residual operators; for the conservative choice $iq_t=q_{xx}\pm 2|q|^2q,\quad r=\mp q^\ast$8, the transfer matrix becomes a Cayley-type update that preserves the quadratic invariant for real $iq_t=q_{xx}\pm 2|q|^2q,\quad r=\mp q^\ast$9 (Medvedev et al., 2019).

Higher-order fast schemes use rational approximations and generalized Cayley transforms. A sixth-order conservative family approximates the step matrix by

$q_t\pm 6q^2q_x+q_{xxx}=0,\quad r=\mp q$0

with the spectral variable mapped by

$q_t\pm 6q^2q_x+q_{xxx}=0,\quad r=\mp q$1

This rational dependence is compatible with fast polynomial algorithms for many spectral values (Medvedev et al., 2020).

Spectral and collocation approaches form a second line of development. A Chebyshev method with the map

$q_t\pm 6q^2q_x+q_{xxx}=0,\quad r=\mp q$2

compresses $q_t\pm 6q^2q_x+q_{xxx}=0,\quad r=\mp q$3 to $q_t\pm 6q^2q_x+q_{xxx}=0,\quad r=\mp q$4, redistributes Chebyshev nodes according to the rapid-changed interval of the potential, and turns the eigenvalue problem into a $q_t\pm 6q^2q_x+q_{xxx}=0,\quad r=\mp q$5 matrix problem solved by the QR algorithm (Cui et al., 2022). For $q_t\pm 6q^2q_x+q_{xxx}=0,\quad r=\mp q$6, $q_t\pm 6q^2q_x+q_{xxx}=0,\quad r=\mp q$7, $q_t\pm 6q^2q_x+q_{xxx}=0,\quad r=\mp q$8, the reported absolute errors were $q_t\pm 6q^2q_x+q_{xxx}=0,\quad r=\mp q$9 and $$w_x=\begin{pmatrix}-ik & q(x)\ r(x) & ik\end{pmatrix}w,\qquad w\in\mathbb C^2,$$00 for the two discrete eigenvalues (Cui et al., 2022).

A third line uses spectral parameter power series. One approach reduces

$$w_x=\begin{pmatrix}-ik & q(x)\ r(x) & ik\end{pmatrix}w,\qquad w\in\mathbb C^2,$$01

to the quadratic Sturm–Liouville pencil

$$w_x=\begin{pmatrix}-ik & q(x)\ r(x) & ik\end{pmatrix}w,\qquad w\in\mathbb C^2,$$02

then constructs convergent SPPS expansions by recursive integrals involving a nonvanishing zero-parameter solution (Kravchenko et al., 2014). A more recent SPPS treatment of direct and inverse scattering introduces

$$w_x=\begin{pmatrix}-ik & q(x)\ r(x) & ik\end{pmatrix}w,\qquad w\in\mathbb C^2,$$03

which maps the upper half-plane to the unit disk. In that variable, the Jost solutions become convergent power series, direct scattering reduces to coefficient computation and root finding in $$w_x=\begin{pmatrix}-ik & q(x)\ r(x) & ik\end{pmatrix}w,\qquad w\in\mathbb C^2,$$04, and inverse scattering reduces to an overdetermined linear algebraic system $$w_x=\begin{pmatrix}-ik & q(x)\ r(x) & ik\end{pmatrix}w,\qquad w\in\mathbb C^2,$$05, with the potential recovered from the first coefficient (Kravchenko, 4 May 2025).

Finally, full scattering-data computation can be organized through structured Volterra systems and Marchenko kernels. In one such pipeline, the direct problem is reformulated in terms of auxiliary kernels $$w_x=\begin{pmatrix}-ik & q(x)\ r(x) & ik\end{pmatrix}w,\qquad w\in\mathbb C^2,$$06, then the continuous and discrete scattering data are extracted via Fourier transforms and matrix-pencil identification of monomial-exponential sums (Fermo et al., 2015). This route computes transmission, left and right reflection coefficients, bound states, multiplicities, and norming constants in a single framework (Fermo et al., 2015).

6. Generalized AKNS systems, gauge equivalence, and recursion operators

The ZS system extends naturally from the $$w_x=\begin{pmatrix}-ik & q(x)\ r(x) & ik\end{pmatrix}w,\qquad w\in\mathbb C^2,$$07 case to Lie-algebraic and matrix AKNS systems. The generalized Zakharov–Shabat operator is

$$w_x=\begin{pmatrix}-ik & q(x)\ r(x) & ik\end{pmatrix}w,\qquad w\in\mathbb C^2,$$08

with $$w_x=\begin{pmatrix}-ik & q(x)\ r(x) & ik\end{pmatrix}w,\qquad w\in\mathbb C^2,$$09 in a Cartan subalgebra and $$w_x=\begin{pmatrix}-ik & q(x)\ r(x) & ik\end{pmatrix}w,\qquad w\in\mathbb C^2,$$10 taking values in the corresponding complement (Grahovski, 2011). For complex regular $$w_x=\begin{pmatrix}-ik & q(x)\ r(x) & ik\end{pmatrix}w,\qquad w\in\mathbb C^2,$$11, the spectral plane is divided into sectors by the rays

$$w_x=\begin{pmatrix}-ik & q(x)\ r(x) & ik\end{pmatrix}w,\qquad w\in\mathbb C^2,$$12

yielding the Caudrey–Beals–Coifman systems and sectorial fundamental analytic solutions rather than the upper/lower half-plane structure of the classical real case (Grahovski, 2011).

Gauge transformation by a fundamental solution at $$w_x=\begin{pmatrix}-ik & q(x)\ r(x) & ik\end{pmatrix}w,\qquad w\in\mathbb C^2,$$13 produces the pole-gauge or Heisenberg-ferromagnet-type form

$$w_x=\begin{pmatrix}-ik & q(x)\ r(x) & ik\end{pmatrix}w,\qquad w\in\mathbb C^2,$$14

This preserves the zero-curvature representation while changing the phase-space variables and Hamiltonian structures (Grahovski, 2011). On $$w_x=\begin{pmatrix}-ik & q(x)\ r(x) & ik\end{pmatrix}w,\qquad w\in\mathbb C^2,$$15, the recursion operators in pole gauge can be written explicitly in terms of the moving Cartan basis $$w_x=\begin{pmatrix}-ik & q(x)\ r(x) & ik\end{pmatrix}w,\qquad w\in\mathbb C^2,$$16, the Gram matrix $$w_x=\begin{pmatrix}-ik & q(x)\ r(x) & ik\end{pmatrix}w,\qquad w\in\mathbb C^2,$$17, the projector $$w_x=\begin{pmatrix}-ik & q(x)\ r(x) & ik\end{pmatrix}w,\qquad w\in\mathbb C^2,$$18, and $$w_x=\begin{pmatrix}-ik & q(x)\ r(x) & ik\end{pmatrix}w,\qquad w\in\mathbb C^2,$$19: $$w_x=\begin{pmatrix}-ik & q(x)\ r(x) & ik\end{pmatrix}w,\qquad w\in\mathbb C^2,$$20 These operators are the duals of Nijenhuis tensors and define Poisson–Nijenhuis structures on the manifold of potentials (Yanovski et al., 2012).

A matrix extension with one zero diagonal entry takes

$$w_x=\begin{pmatrix}-ik & q(x)\ r(x) & ik\end{pmatrix}w,\qquad w\in\mathbb C^2,$$21

with $$w_x=\begin{pmatrix}-ik & q(x)\ r(x) & ik\end{pmatrix}w,\qquad w\in\mathbb C^2,$$22 anticommuting with $$w_x=\begin{pmatrix}-ik & q(x)\ r(x) & ik\end{pmatrix}w,\qquad w\in\mathbb C^2,$$23 (Mee, 19 Nov 2025). The presence of the zero entry creates a missing Jost solution that is recovered through the dual system and wedge products in exterior algebra. With that device, the direct and inverse scattering theory, corrected transmission and reflection coefficients, Wiener–Hopf factorization, Marchenko equations, and time evolution all extend to this singular AKNS setting (Mee, 19 Nov 2025).

7. Geometric, probabilistic, and field-theoretic extensions

The ZS system is not confined to soliton PDEs. In the weak-noise theory of the one-dimensional KPZ equation at short time, the saddle-point system

$$w_x=\begin{pmatrix}-ik & q(x)\ r(x) & ik\end{pmatrix}w,\qquad w\in\mathbb C^2,$$24

admits an AKNS/ZS Lax pair

$$w_x=\begin{pmatrix}-ik & q(x)\ r(x) & ik\end{pmatrix}w,\qquad w\in\mathbb C^2,$$25

with

$$w_x=\begin{pmatrix}-ik & q(x)\ r(x) & ik\end{pmatrix}w,\qquad w\in\mathbb C^2,$$26

The inverse scattering transform then gives exact formulas for $$w_x=\begin{pmatrix}-ik & q(x)\ r(x) & ik\end{pmatrix}w,\qquad w\in\mathbb C^2,$$27 and $$w_x=\begin{pmatrix}-ik & q(x)\ r(x) & ik\end{pmatrix}w,\qquad w\in\mathbb C^2,$$28 through Fredholm operators, and for the droplet initial condition yields explicit large-deviation functions and a solitonic second branch generated when the scattering data develop a pole (Krajenbrink et al., 2021).

In integrable probability, the largest real eigenvalue of the real Ginibre ensemble is governed by a distinguished ZS inverse-scattering problem with Gaussian reflection coefficient

$$w_x=\begin{pmatrix}-ik & q(x)\ r(x) & ik\end{pmatrix}w,\qquad w\in\mathbb C^2,$$29

The associated Riemann–Hilbert problem gives a ZS system

$$w_x=\begin{pmatrix}-ik & q(x)\ r(x) & ik\end{pmatrix}w,\qquad w\in\mathbb C^2,$$30

and leads to exact closed-form and Fredholm-determinant representations for the limiting distribution (Baik et al., 2018). A related Fredholm-determinant hierarchy $$w_x=\begin{pmatrix}-ik & q(x)\ r(x) & ik\end{pmatrix}w,\qquad w\in\mathbb C^2,$$31 yields an explicit unique solution of the inverse scattering transform for the ZS system through the matrix formula

$$w_x=\begin{pmatrix}-ik & q(x)\ r(x) & ik\end{pmatrix}w,\qquad w\in\mathbb C^2,$$32

thereby linking ZS inverse scattering to the real Ginibre problem and to a broader quasi-universal determinantal hierarchy (Krajenbrink, 2020).

A geometric application uses the Zakharov–Shabat dressing method to generate exact solutions of the minimal-surface equation

$$w_x=\begin{pmatrix}-ik & q(x)\ r(x) & ik\end{pmatrix}w,\qquad w\in\mathbb C^2,$$33

in $$w_x=\begin{pmatrix}-ik & q(x)\ r(x) & ik\end{pmatrix}w,\qquad w\in\mathbb C^2,$$34 by embedding it into a $$w_x=\begin{pmatrix}-ik & q(x)\ r(x) & ik\end{pmatrix}w,\qquad w\in\mathbb C^2,$$35 Lax representation and reconstructing the dressed surface by path-independent quadrature (Gutshabash, 2014). In that setting the dressing factor is normalized by $$w_x=\begin{pmatrix}-ik & q(x)\ r(x) & ik\end{pmatrix}w,\qquad w\in\mathbb C^2,$$36, the dressed field is reconstructed from $$w_x=\begin{pmatrix}-ik & q(x)\ r(x) & ik\end{pmatrix}w,\qquad w\in\mathbb C^2,$$37, and nontrivial deformations of the helicoid are obtained from the same algebraic mechanism that underlies inverse scattering (Gutshabash, 2014).

Across these extensions, the recurring role of the ZS system is structural rather than merely notational: it supplies the spectral problem, the analytic continuation, the scattering data, and the inverse map. Whether the output is an $$w_x=\begin{pmatrix}-ik & q(x)\ r(x) & ik\end{pmatrix}w,\qquad w\in\mathbb C^2,$$38-soliton solution, a Bohr–Sommerfeld quantization rule, a Fredholm determinant, a KPZ rate function, or a dressed minimal surface, the same first-order $$w_x=\begin{pmatrix}-ik & q(x)\ r(x) & ik\end{pmatrix}w,\qquad w\in\mathbb C^2,$$39 or AKNS-type spectral architecture remains the organizing principle (Yagasaki, 8 Jun 2025, Dong et al., 8 May 2025, Krajenbrink et al., 2021, Baik et al., 2018).