2+1D S-Integrable Extended Dym Equation

• The paper introduces a 2+1D S-integrable extended Dym-type equation that generalizes classical integrable models via a novel Lax pair and ⁠𝜕̄-dressing scheme.
• It employs reciprocal-gauge and Ermakov-type transformations to derive spatially and temporally modulated yet integrable systems.
• Key implications include applications in geometric flows and hydrodynamics, offering a framework for modeling nonlinear, inhomogeneous dynamics.

The 2+1-dimensional S-integrable extended Dym-type equation is a nonlinear partial differential equation generalizing the integrable Dym equation to two spatial and one temporal dimension. This equation, introduced by Konopelchenko, Rogers, and Amster, possesses rich integrable structure, admits a Lax pair, and supports a $\bar\partial$-dressing scheme. Its modulated versions emerge through a class of involutory transformations rooted in Ermakov theory, permitting spatially, temporally, or dynamically inhomogeneous integrable systems. The equation is significant in the context of geometric flows, hydrodynamics, and the theory of integrable modulation.

1. Definition and Canonical Formulation

Let $u = u(x, y, t)$ be a real scalar field. The S-integrable 2+1-dimensional extended Dym-type equation is given by

$$u_{t} + 2\,\partial_x(1-\partial_{xx})(u^{-\tfrac12}) + 6\,u^2 \left[ u^{-1}\partial_x^{-1}(u^{\tfrac12})_y \right]_y = 0, \tag{E1}$$

where $\partial_x = \frac{\partial}{\partial x}$, $\partial_x^{-1}f(x) \equiv \int^x f(s)\,ds$, and analogous definitions for $\partial_y$, $\partial_t$. Introducing the transformation $u = 1/V^2$, the system becomes

$V_t = V^3(V_x - V_{xxx}) + \frac{3}{V} (V^2 \rho)_y, \quad V_y = -V^2 \rho_x, \tag{E1%%%%7%%%%}$

with $\rho$ an auxiliary potential. Form (E1$u = u(x, y, t)$0) is preferred for linear representations and establishing integrability.

2. Lax Pair Representation

Integrability is established via the existence of a Lax pair. Introducing the wave function $u = u(x, y, t)$1 and spectral parameter $u = u(x, y, t)$2, the Lax pair reads: $u = u(x, y, t)$3 with auxiliary relations $u = u(x, y, t)$4, $u = u(x, y, t)$5. The compatibility condition $u = u(x, y, t)$6 is equivalent to the dynamical system (E1$u = u(x, y, t)$7), thus establishing S-integrability (Konopelchenko et al., 9 Jan 2026).

3. $u = u(x, y, t)$8-Dressing Scheme

The analytical construction of solutions exploits the $u = u(x, y, t)$9-dressing method, utilizing a nonlocal problem in the complex $$u_{t} + 2\,\partial_x(1-\partial_{xx})(u^{-\tfrac12}) + 6\,u^2 \left[ u^{-1}\partial_x^{-1}(u^{\tfrac12})_y \right]_y = 0, \tag{E1}$$0-plane: $$u_{t} + 2\,\partial_x(1-\partial_{xx})(u^{-\tfrac12}) + 6\,u^2 \left[ u^{-1}\partial_x^{-1}(u^{\tfrac12})_y \right]_y = 0, \tag{E1}$$1 normalized with $$u_{t} + 2\,\partial_x(1-\partial_{xx})(u^{-\tfrac12}) + 6\,u^2 \left[ u^{-1}\partial_x^{-1}(u^{\tfrac12})_y \right]_y = 0, \tag{E1}$$2 as $$u_{t} + 2\,\partial_x(1-\partial_{xx})(u^{-\tfrac12}) + 6\,u^2 \left[ u^{-1}\partial_x^{-1}(u^{\tfrac12})_y \right]_y = 0, \tag{E1}$$3. The kernel $$u_{t} + 2\,\partial_x(1-\partial_{xx})(u^{-\tfrac12}) + 6\,u^2 \left[ u^{-1}\partial_x^{-1}(u^{\tfrac12})_y \right]_y = 0, \tag{E1}$$4 depends on a bare phase,

$$u_{t} + 2\,\partial_x(1-\partial_{xx})(u^{-\tfrac12}) + 6\,u^2 \left[ u^{-1}\partial_x^{-1}(u^{\tfrac12})_y \right]_y = 0, \tag{E1}$$5

with $$u_{t} + 2\,\partial_x(1-\partial_{xx})(u^{-\tfrac12}) + 6\,u^2 \left[ u^{-1}\partial_x^{-1}(u^{\tfrac12})_y \right]_y = 0, \tag{E1}$$6. Dressed operators

$$u_{t} + 2\,\partial_x(1-\partial_{xx})(u^{-\tfrac12}) + 6\,u^2 \left[ u^{-1}\partial_x^{-1}(u^{\tfrac12})_y \right]_y = 0, \tag{E1}$$7

allow construction of Lax-like equations for $$u_{t} + 2\,\partial_x(1-\partial_{xx})(u^{-\tfrac12}) + 6\,u^2 \left[ u^{-1}\partial_x^{-1}(u^{\tfrac12})_y \right]_y = 0, \tag{E1}$$8. Imposing the vanishing residue condition at $$u_{t} + 2\,\partial_x(1-\partial_{xx})(u^{-\tfrac12}) + 6\,u^2 \left[ u^{-1}\partial_x^{-1}(u^{\tfrac12})_y \right]_y = 0, \tag{E1}$$9 yields the algebraic constraints $\partial_x = \frac{\partial}{\partial x}$0, $\partial_x = \frac{\partial}{\partial x}$1, and a system of PDEs for $\partial_x = \frac{\partial}{\partial x}$2, whose elimination recovers a potential form of (E1). Every solution of the $\partial_x = \frac{\partial}{\partial x}$3-problem yields, via $\partial_x = \frac{\partial}{\partial x}$4 and $\partial_x = \frac{\partial}{\partial x}$5, a solution $\partial_x = \frac{\partial}{\partial x}$6 to the extended Dym equation (Konopelchenko et al., 9 Jan 2026).

4. Involutory (Ermakov-Type) Transformations

The generalized Dym equations admit involutory (self-inverse) transformations—both spatial and temporal—serving as reciprocal-gauge deformations. For the canonical S-integrable equations these take the form: $\partial_x = \frac{\partial}{\partial x}$7 with $\partial_x = \frac{\partial}{\partial x}$8. For temporal rescaling: $\partial_x = \frac{\partial}{\partial x}$9 with $\partial_x^{-1}f(x) \equiv \int^x f(s)\,ds$0. These involutory transformations generate hierarchies of modulated integrable equations, allowing the construction of spatially and temporally inhomogeneous yet integrable Dym-type systems (Konopelchenko et al., 9 Jan 2026).

5. Integrable Modulated Equations

Spatial and temporal modulations generalize the homogeneous equation while preserving integrability:

• Spatial modulation: Application of $\partial_x^{-1}f(x) \equiv \int^x f(s)\,ds$1 to the canonical Dym equation $\partial_x^{-1}f(x) \equiv \int^x f(s)\,ds$2 yields a modulated equation involving variable coefficients $\partial_x^{-1}f(x) \equiv \int^x f(s)\,ds$3, with the modulated Lax pair derivable by conjugation.
• Temporal modulation: Modulating the extended Dym field $\partial_x^{-1}f(x) \equiv \int^x f(s)\,ds$4 by $\partial_x^{-1}f(x) \equiv \int^x f(s)\,ds$5,

$\partial_x^{-1}f(x) \equiv \int^x f(s)\,ds$6

Setting $\partial_x^{-1}f(x) \equiv \int^x f(s)\,ds$7 recovers the unmodulated equation (E1).

• Ermakov modulation: Imposing that $\partial_x^{-1}f(x) \equiv \int^x f(s)\,ds$8 obey the classical Ermakov–Pinney equation,

$\partial_x^{-1}f(x) \equiv \int^x f(s)\,ds$9

with general solution

$\partial_y$0

leads to a time-modulated, integrable equation

$\partial_y$1

Integrability of these modulated systems is inherited via Lax pair conjugation (Konopelchenko et al., 9 Jan 2026).

6. Illustrative Special Cases and Structural Properties

Selected examples elucidate the flexibility enabled by modulation:

• Constant modulation: $\partial_y$2 or $\partial_y$3 constant reduces all deformed systems to their canonical Dym or extended Dym forms.
• Linear spatial weights: $\partial_y$4 models power-law spatial inhomogeneity, relevant for media with variable stiffness or density.
• Oscillatory Ermakov modulation: With $\partial_y$5, the Ermakov equation solutions are trigonometric, resulting in $\partial_y$6 being a periodic prefactor. This models time-periodic parametric effects, e.g., pumping or oscillatory shells in physical systems.

For each case, the associated Lax operators are obtained by replacing derivatives and rescaling potentials by the reciprocal/gauge-transformed variables. Integrability is preserved, with the zero-curvature (zero commutator) criterion and conservation law structure retained, ensuring the existence of infinitely many conserved quantities and exact solution methods (Konopelchenko et al., 9 Jan 2026).

7. Context and Significance in Mathematical Physics

The S-integrable 2+1-dimensional extended Dym-type equation generalizes the well-known Dym and Camassa–Holm-type hierarchies to higher dimensions and inhomogeneous settings. Its connections with classical geometric flow models, hydrodynamical peakon dynamics, and representation through Ermakov-type transformations place it at the intersection of soliton theory, symmetry reduction techniques, and mathematical physics. The systematic construction of modulated, yet fully integrable systems offers a mechanism for encoding spatial and temporal inhomogeneities in exact models, thus expanding the toolkit available for both theoretical analysis and physical applications in contexts where such inhomogeneities are essential (Konopelchenko et al., 9 Jan 2026).