Integrable Modulated 2+1 Extended Dym Eq

• The integrable modulated extended Dym equation generalizes solitonic dynamics by incorporating involutory Ermakov-type transformations for spatial and temporal modulation.
• Lax pair formalism and 𝛛̅-dressing techniques ensure the preservation of integrable structure while handling nonuniform, time-varying media.
• Explicit Ermakov modulations offer analytic templates to model variable dispersion in physical settings like graded wave-guides and cold-plasma oscillations.

The integrable modulated versions of the 2+1-dimensional extended Dym equation generalize classical solitonic dynamics by introducing modulated media via involutory transformations rooted in Ermakov theory. Building upon the canonical Dym equation's appearance in geometric torsion evolution and hydrodynamic peakon phenomena, these extensions exploit both spatial and temporal modulation, leading to nonuniform and oscillator-controlled wave propagation while preserving integrable structure through transform-induced Lax pairs and $\bar\partial$-dressing techniques (Konopelchenko et al., 9 Jan 2026).

1. The 2+1-Dimensional Extended Dym Equation

The fundamental object is the extended Dym-type equation for a strictly positive field $u=u(x,y,t)$, defined over continuous spatial variables $x, y \in \mathbb{R}$ and temporal $t \in \mathbb{R}$:

$$u_t \;+\;2\,\partial_x\bigl(1-\partial_{xx}\bigr)\Bigl(\tfrac{1}{\sqrt u}\Bigr) \;+\;6\,u^2\Bigl[u^{-1}\,\partial_x^{-1}\bigl(\sqrt u\bigr)_y\Bigr]_y \;=\;0\,. \tag{E}$$

This equation can equivalently be reformulated by setting $u=1/V^2$ for $V(x,y,t)>0$, introducing an auxiliary field $\rho(x,y,t)$ governed by:

$$\begin{cases} V_t = V^3 (V_x - V_{xxx}) + \dfrac{3}{V} (V^2 \rho)_y,\ V_y = -V^2 \rho_x, \end{cases} \qquad \rho_x - (1/V)_y = 0.$$

This system is fully equivalent to the extended Dym equation (E) and serves as the basis for further integrable analysis.

2. Lax Pair Formalism and Integrability Structure

Integrability of the extended Dym equation is established via a two-component scalar Lax pair. Introducing a wave function $\phi(x,y,t;\lambda)$ dependent on the complex spectral parameter $\lambda$, one considers the overdetermined system:

$\begin{cases} L_1(\lambda)\,\phi = \phi_y + V^2(\phi_{xx}+\phi_x) = 0,\[1ex] L_2(\lambda)\,\phi = \phi_t + 4\,V^3\,\phi_{xxx} + 6\,V^3\,\phi_{xx} + 2\,V^3\,\phi_x + 6\,V^2\,(V_x+\rho)\,(\phi_{xx}+\phi_x) = 0. \end{cases} \tag{L}$

The spectral parameter $\lambda$ enters only parametrically; there is no differentiation with respect to $\lambda$. The compatibility (zero-curvature) condition $[L_1, L_2]\phi = 0$ is exactly equivalent to the system for $(V, \rho)$ and thus for $u(x,y,t)$, demonstrating that the equation admits an S-integrable structure via this Lax pair representation.

3. $\bar{\partial}$-Dressing Construction

The solution space and integrability properties are further elucidated using a $\bar{\partial}$-dressing scheme, which constructs solutions via a nonlocal integral equation for a "dressing function" $\chi(\lambda,\bar\lambda;x,y,t)$:

$$\frac{\partial\chi}{\partial\bar\lambda} = \iint_{\mathbb C} \chi(\lambda',\bar\lambda')\, R(\lambda',\bar\lambda';\lambda,\bar\lambda;\,x,y,t)\, d\lambda'\wedge d\bar\lambda'\,, \tag{D}$$

with $\chi\to1$ as $|\lambda|\to\infty$. The kernel $R$ involves a phase $F$:

$$F(\lambda;x,y,t) =\frac1\lambda\,f(x,y,t) +\frac1{\lambda^2}\,y +\frac{4\,i}{\lambda^3}\,t,$$

and the auxiliary function $f(x,y,t)$ is subject to determining equations arising from closure under operator expansions:

$$D_x=\partial_x+\frac{i}{\lambda}f_x,\quad D_y=\partial_y+\frac{1}{\lambda^2}+\frac{i}{\lambda}f_y,\quad D_t=\partial_t+\frac{4\,i}{\lambda^3}+\frac{i}{\lambda}f_t.$$

Operators for compatibility:

$$L_1 = D_y+V^2(D_x^2+D_x),\qquad L_2 = D_t+4V^3D_x^3+6V^3D_x^2+2V^3D_x +6V^2(V_x+\rho)(D_x^2+D_x).$$

Imposing analyticity constraints yields $V=1/f_x$ and $\rho=f_y$, with $f$ satisfying the potential form of the 2+1 extended Dym equation:

$$f_{tx}-6f_y f_{xy}+3f_x f_{yy} +\frac{f_{xxxx}}{f_x^3} -6\frac{f_{xx}f_{xxx}}{f_x^4} -\frac{f_{xx}}{f_x^3} +6\frac{f_{xx}^3}{f_x^5} =0.$$

The field reconstruction $V=1/f_x$, $\rho=f_y$ with the solution $\phi=\chi\,e^F$ recovers the Lax pair solution structure.

4. Involutory (Ermakov-Type) Modulation Transformations

A central mechanism for generating modulated versions is the application of special involutory transformations which conjugate the canonical equation into modulated forms:

(a) Spatial Modulation $\mathcal R^*$

For spatial modulation, define:

$$x^*=\int^x\rho^{-2}(s)ds,\qquad r^*=\rho^{-1}(x)r, \qquad y^*=y,\;t^*=t,\;\rho^*=\rho^{-1}(x).$$

This transformation carries the Dym equation to a spatially modulated version with nonuniform derivatives in $x^*$. Double application restores the original variables, establishing involutivity.

(b) Temporal Modulation $\mathcal I^*$

Temporal modulation is defined by:

$$t^* = \int^t \rho^{-2}(s)ds, \quad x^*=x,\,y^*=y, \quad u^*(x^*,y^*,t^*) = \frac{u(x,y,t)}{\rho(t)}, \quad \rho^*(t^*) = \frac{1}{\rho(t)}.$$

Applying $\mathcal I^*$ twice yields the original setting. The resulting temporally modulated equation is:

$$\partial_{t^*}(\rho^{*-1}u^*) + 2\,\rho^{*-3/2} \partial_{x^*}(1-\partial_{x^*x^*})\left(\frac{1}{\sqrt{u^*}}\right) +6\,\rho^{*-5/2} u^{*2} [u^{*-1} \partial_{x^*}^{-1}(\sqrt{u^*})_{y^*}]_{y^*} = 0. \tag{TM}$$

5. Explicit Ermakov Modulations and Their Solutions

The temporal modulation parameter $\rho^*(t^*)$ can be chosen to solve a classical Ermakov equation:

$$\rho^*_{t^*t^*} + w(t^*) \rho^* = \frac{\mathcal E}{(\rho^*)^3},\quad \mathcal E \in \mathbb{R}. \tag{Er}$$

Its general solution is constructed from two linearly independent functions $\Omega_1, \Omega_2$ satisfying

$\Omega_{t^* t^*} + w(t^*) \Omega = 0,$

with

$$\rho^*(t^*) = (c_1 \Omega_1^2 + 2c_2 \Omega_1 \Omega_2 + c_3 \Omega_2^2)^{1/2},\qquad c_1c_3 - c_2^2 = \mathcal E/W^2,$$

$W$ being the Wronskian $W = \Omega_1 \dot\Omega_2 - \dot\Omega_1 \Omega_2$. Substitution of this profile into (TM) yields the Ermakov-modulated equation:

$\begin{split} \frac{\partial}{\partial t^*} \left[ (c_1\Omega_1^2 + 2c_2\Omega_1\Omega_2 + c_3\Omega_2^2)^{1/2} u^* \right] &+ 2(c_1\Omega_1^2 + 2c_2\Omega_1\Omega_2 + c_3\Omega_2^2)^{3/4} \ &\times \partial_{x^*}(1-\partial_{x^* x^*})(u^{*-1/2}) \ &+ 6(c_1\Omega_1^2 + 2c_2\Omega_1\Omega_2 + c_3\Omega_2^2)^{5/4} u^{*2} [u^{*-1} \partial_{x^*}(u^{*1/2})_{y^*}]_{y^*} = 0. \end{split} \tag{EM}$

Involution of the transformation and preservation of Lax integrability ensures integrable structure for these modulated equations.

6. Illustrative Modulation Cases and Physical Interpretation

Several specific modulations highlight the utility and range of this framework:

• Trivial modulation: $\rho^*(t^*) \equiv 1$ recovers the canonical extended Dym equation (E).
• Harmonic Ermakov modulation: $w(t^*) = \omega^2 > 0,\, \mathcal E = 0$, with $\Omega_1 = \cos(\omega t^*)$, $\Omega_2 = \sin(\omega t^*)$ yields

$$\rho^*(t^*) = \sqrt{c_1 \cos^2(\omega t^*) + 2c_2 \sin(\omega t^*) \cos(\omega t^*) + c_3 \sin^2(\omega t^*)}$$

describing an extended Dym-type wave in an oscillatorily time-varying medium.

• Spatial exponential modulation: For $r(x,y,t)$, take $\rho(x) = e^{k x}$, so $x^* = (1/k) e^{-k x}$, $r^* = e^{-k x} r$. This constructs a Dym equation with spatially graded inhomogeneous coefficients.

Physical interpretations relate spatial modulation to nonuniform media such as graded index wave-guides, while temporal modulation via Ermakov profiles model variable-dispersion environments relevant to thin-shell hyperelastic dynamics and cold-plasma oscillations.

7. Mathematical and Physical Significance

The integrable modulated forms of the 2+1-dimensional extended Dym equation encompass a novel class of S-integrable systems anchored in the formal machinery of Lax pairs and $\bar\partial$-dressing. Through involutory Ermakov-type transformations, such systems admit explicit handling of nonuniform and time-varying environments without sacrificing integrable solvability. This suggests broader applicability in modeling complex physical systems where medium properties evolve according to oscillator laws or spatial grading, and provides concrete analytic templates for the study of integrable equations in modulated contexts (Konopelchenko et al., 9 Jan 2026).