(2+1)D Damping-Forcing Coupled Burgers Eq.

• The paper presents a (2+1)D coupled Burgers equation with time-dependent damping and forcing, enabling the generation of explicit solitary and periodic solutions via Darboux transformations.
• It establishes integrability by constructing a Lax pair and employing the zero-curvature formalism, which rigorously links the governing coupled PDEs.
• The study shows that tuning the damping and forcing functions directly affects the amplitude, stability, and background of the wave solutions.

The (2+1)-dimensional damping-forcing coupled Burgers equation (DFCB) generalizes the classical Burgers system to two spatial dimensions and includes explicit time-dependent damping and forcing terms. This system models two real fields, $u(x,y,t)$ and $v(x,y,t)$, with coupled nonlinear PDEs whose coefficients are governed by arbitrary functions $\Lambda(t)$ and $H(t)$. The integrability of the DFCB equation permits the construction of its Lax pair, enabling the application of Darboux transformations to generate explicit solitary and periodic wave solutions. The temporal profiles of damping and forcing critically influence the amplitude, stability, and background of these solutions, affording rich dynamics modulated by function parameters (Chatterjee et al., 20 Jan 2026).

1. Formulation and Parameterization of the DFCB System

The DFCB equation comprises the following coupled PDEs for $u(x,y,t)$ and $v(x,y,t)$: $$\begin{align*} u_t - 2u_y + \frac{3}{2}u_{xx} - 3v_x - S(t)\,u &= T(t), \ v_t - 2v_y - u_y + u_{xxx} + P(t)\,u\,u_x - \frac{3}{2}v_{xx} + R(t)\,u_x - S(t)\,v &= T(t), \end{align*}$$ where the coefficients are defined via

$$\begin{align*} P(t) &= \frac{1}{\Lambda(t)}, & R(t) &= -\frac{H(t)}{\Lambda(t)}, \ S(t) &= \frac{\Lambda'(t)}{\Lambda(t)}, & T(t) &= H'(t) - \frac{\Lambda'(t)}{\Lambda(t)}H(t). \end{align*}$$

Here, $S(t)$ implements linear damping, and $T(t)$ represents a uniform external source. The equation recovers the standard coupled Burgers system in two spatial dimensions for the choice $\Lambda \equiv 1, H \equiv 0$.

2. Lax Integrability and Zero-Curvature Formalism

Integrability of the DFCB system is established via the Lax pair for a scalar eigenfunction $\psi(x,y,t)$: $L\psi = \psi_y, \qquad \psi_t = M\psi,$ where

$$\begin{align*} L &= \partial_x^3 + \frac{u-H(t)}{\Lambda(t)}\,\partial_x + \frac{v-H(t)}{\Lambda(t)}\,I, \ M &= \partial_x^3 + \frac{3}{2}\partial_x^2 + \frac{u-H(t)}{\Lambda(t)}\,\partial_x + \partial_y + \frac{u+v-2H(t)}{\Lambda(t)}\,I. \end{align*}$$

$I$ denotes the identity operator. The DFCB equations follow from the zero-curvature condition $L_t - M_y + [L, M] = 0$, confirming their integrable structure.

3. General $N$-Fold Darboux Transformation

Given $N$ linearly independent solutions $\psi_1, \dots, \psi_N$ of the Lax pair, the $N$-fold Darboux transformation produces modified potentials $u^{[N]}, v^{[N]}$: $$\begin{align*} \psi^{[N]} &= \frac{W(\psi_1, \dots, \psi_N, \psi)}{W(\psi_1, \dots, \psi_N)}, \ u^{[N]} &= u + 3\Lambda(t)\,\partial_x^2\ln W(\psi_1, \dots, \psi_N), \ v^{[N]} &= v + N u_x + 3\Lambda(t) \sum_{r=1}^N \left[ \partial_x \ln \frac{W(\psi_1, \dots, \psi_r)}{W(\psi_1, \dots, \psi_{r-1})}\, \partial_x^2 \ln \frac{W(\psi_1, \dots, \psi_r)}{W(\psi_1, \dots, \psi_{r-1})} + \partial_x^3\ln W(\psi_1, \dots, \psi_r) \right], \end{align*}$$ where $W(\cdot)$ is the Wronskian determinant in $x$. This transformation recursively generates exact solutions by modulating the spectral content and allows for diverse waveforms, including solitary and periodic types.

4. Explicit Construction: One-Fold and Two-Fold Darboux Transformations

One-Fold Case

On the trivial background $u(x,y,t) = H(t)$, $v(x,y,t) = H(t)$, a seed solution is: $$\psi_1(x, y, t) = c_1 e^{\xi_1} + c_2 e^{\xi_2} \sin \xi_3 + c_3 e^{\xi_2} \cos \xi_3,$$ with

$$\begin{align*} \xi_1 &= k_1 x + k_1^3 y + (2k_1^3 + \frac{3}{2}k_1^2)t, \ \xi_2 &= k_1 x - 2k_1^3 y - 4k_1^3 t, \ \xi_3 &= k_1 x + 2k_1^3 y + (4k_1^3 + 3k_1^2)t. \end{align*}$$

Applying the one-fold Darboux transform: $$\begin{align*} u^{[1]} &= H(t) + 3\Lambda(t)\,\partial_x^2\ln\psi_1, \ v^{[1]} &= H(t) + 3\Lambda(t) \left[ \partial_x\ln\psi_1\,\partial_x^2\ln\psi_1 + \partial_x^3\ln\psi_1 \right], \end{align*}$$ produces explicit solitary or periodic waveforms via parameter choices.

Two-Fold Case

With two seeds,

$$\psi_2(x, y, t) = p_1 e^{\delta_1} + p_2 e^{\delta_2} \sin \delta_3 + p_3 e^{\delta_2} \cos \delta_3,$$

($\delta_i$ as above, with $k_2$), the Wronskian

$$W_{12}(x) = \begin{vmatrix} \psi_1 & \psi_2 \ \partial_x\psi_1 & \partial_x\psi_2 \end{vmatrix}$$

generates

$$\psi^{[2]} = \frac{W(\psi_1, \psi_2, \psi)}{W_{12}},$$

with

$$\begin{align*} u^{[2]} &= H(t) + 3\Lambda(t)\,\partial_x^2\ln W_{12}, \ v^{[2]} &= H(t) + 2H'(t) + 3\Lambda(t) \bigl[ (\partial_x\ln\psi_1)\partial_x^2\ln\psi_1 + \partial_x^3\ln\psi_1 + (\partial_x\ln\frac{W_{12}}{\psi_1})\partial_x^2\ln\frac{W_{12}}{\psi_1} + \partial_x^3\ln W_{12} \bigr]. \end{align*}$$

This enables the explicit construction of multi-soliton, breather, or periodic solutions.

5. Characteristic Wave Solutions

Solitary wave (one-soliton) profiles arise for exponential seeds, e.g., $\psi_1 = c_1 e^{\xi_1}$: $$u^{[1]}(x, y, t) = H(t) - \frac{3\Lambda(t) k_1^2}{c_1 e^{\xi_1}},$$ embodying the classic $\operatorname{sech}^2$ soliton traveling in the $(x, y)$-plane with velocity determined by $k_1$.

Periodic waves are obtained with trigonometric seed combinations: $$u^{[1]}(x, y, t) = H(t) + 3\Lambda(t) k_1^2 \frac{\cos\xi_3 - \sin\xi_3}{\cos\xi_3 + \sin\xi_3},$$ exhibiting cnoidal-type oscillations whose amplitude is modulated by $\Lambda(t)$.

Two-fold Darboux transformations yield elastic soliton interactions, periodic patterns, or breathers, subject to parameter selection in the seeds.

6. Modulation by Damping and Forcing Functions

The amplitude and background level of the Darboux-generated solutions are governed by the time-dependent coefficients:

• Damping ($S(t)$): As $\Lambda(t)$ appears in the amplitude scaling, its decay (i.e., positive damping) attenuates the amplitude of both solitary and periodic waves over time. The soliton peak amplitude scales like $\Lambda(t)k_1^2$.
• Forcing ($T(t)$): The background $H(t)$, toward which $u^{[N]}$ asymptotes spatially, can grow or decay depending on $T(t)$. If $H(t)$ increases, solitary waves propagate atop a rising background, and for large positive $T(t)$ the system may become unstable.
• Stability analysis by linearization indicates that damping introduces a negative real part to the dispersion relation, which stabilizes short waves. Forcing serves as an external energy input, potentially destabilizing the mean flow if $T(t)$ is sufficiently positive.

7. Graphical Analysis and Physical Implications

Graphical representations include:

• Time evolution of solitary wave profiles $u^{[1]}(x, 0, t)$ for exponentially decaying $\Lambda(t)$, showing decay in peak amplitude and pulse broadening.
• Periodic solutions for $\Lambda(t) = 1$, $H(t) = 0$, displaying unchanging cnoidal waves.
• Elastic collisions of two-soliton profiles $u^{[2]}(x, 0, t)$, where post-collision amplitudes are scaled by $\Lambda(t)$.

This suggests that the explicit Darboux-transformed solutions afford tunable control over shape, amplitude, and background via $\Lambda(t)$ and $H(t)$, enriching the dynamical repertoire of the (2+1)-dimensional coupled Burgers system under damping and forcing (Chatterjee et al., 20 Jan 2026).