A nonlocal coupled modified complex integrable dispersionless equation: Darboux transformation, soliton-type solutions and its asymptotic behavior

TL;DR

This work develops a Darboux transformation framework for the nonlocal cm-CID equation, establishing a Lax pair and symmetry structure to generate N-fold transformations that produce new potentials from a seed. Under both vanishing and non-vanishing boundary conditions, the authors construct a rich zoo of exact solutions, including one- and two-soliton states, periodic, breather-like, and periodic-like waves, as well as rational and rogue-wave-type solutions. A key finding is that the nonlocal cm-CID equation exhibits novel solution types not present in the local cm-CID case, such as growing- and decaying-periodic waves and mixed periodic-like structures, with asymptotic analysis confirming elastic collision behavior in multi-soliton interactions. The results have potential implications for optical pulse propagation in nonlinear fibers and open avenues for further integrability studies, numerical discretization, and inverse-scattering treatments of the nonlocal system.

Abstract

In this paper, we primarily construct Darboux transformation(DT) of the nonlocal coupled modified complex integrable dispersionless (cm-CID) equation, which is first proposed by the connection with a nonlocal coupled modified complex short pulse(cm-CSP) equation. Utilizing DT, we present soliton-type solutions for the nonlocal cm-CID equation under vanishing and non-vanishing boundary conditions. Soliton-type solutions include periodic wave, growing-, decaying-periodic wave, periodic-like wave (which consists of a mixture of periodic wave and breather wave, a combination of periodic wave and background plane), breather-like wave and rational solution. Furthermore, we have also analyzed asymptotic behavior and properties of these solutions theoretically and graphically. We must emphasis that soliton solutions of the nonlocal cm-CID equation possess novel properties that are distinct from those of the cm-CID equation, such as the nonlocal cm-CID equation has the growing-, decaying-periodic solution and periodic-like solution. The implications of these findings could potentially contribute to the description of optical pulse behavior during propagation in optical fibers.

Figures (13)

• Figure 1: Periodic wave solution for the f-f, f-def and def-def nonlocal cm-CID equation with $\lambda_1=\text{i}$.
• Figure 2: Decaying-, growing- and decaying-growing periodic solution for the nonlocal cm-CID equation: $(a)$$c_4=0$, $\lambda_1=\frac{1}{20}+\text{i}$, $(b)$$c_4=0$, $\lambda_1=2+\text{i}$, $(c)$$c_4=\frac{1}{3}$, $\lambda_1=\frac{1}{20}+\text{i}$$(d)$$c_4=\frac{1}{3}$, $\lambda_1=2+\text{i}$.
• Figure 3: Double periodic solution for the nonlocal f-def cm-CID equation with $\lambda_1=\frac{\text{i}}{2},\lambda_2=-\text{i}$.
• Figure 4: Periodic-like wave solution for the f-def nonlocal cm-CID equation with $\alpha_1=\frac{3}{5},\beta_1=\frac{4}{5},\alpha_2=0,\beta_2=-1$: $(a)$-$(c)$$c_3=0,c_7=2$, $(d)$-$(f)$$c_3=-1,c_7=2$, $(a)(d)$ periodic-like waves $|u^{(2)}|$, $(b)(e)$ periodic-like waves $|v^{(2)}|$, $(c)(f)$ breather-like wave $|\rho^{(2)}|$.
• Figure 5: Breather-like solution for the f-def nonlocal cm-CID equation with $c_3=\frac{3}{2},c_7=1,\alpha_1=\frac{1}{2},\beta_1=1,\alpha_2=0,\beta_2=-1$: $(a)(d)$ bright breather-like soliton $|u^{(2)}|$, $(b)(e)$ bright breather-like soliton $|v^{(2)}|$, $(c)(f)$ dark breather-like soliton $|\rho^{(2)}|$.