Soliton,breathers,positons and rogue waves for the vector complex modified Korteweg-de Vries equation

TL;DR

This work develops a determinant-based $N$-fold Darboux transformation for the vector complex modified Korteweg–de Vries equation, enabling the construction of a broad family of globally bounded multi-component waves. By exploiting the Lax pair and DT framework, the authors obtain $N$-fold solitons (bright-bright-bright and dark-bright-bright), $N$-fold breathers on nonvanishing backgrounds, and degenerate-eigenvalue limits that yield $N$-positons and $N$-th order rogue waves, with explicit determinant representations and asymptotic analyses. Most solutions remain bounded, and the paper provides extensive graphical demonstrations of soliton–breather–positon–rogue wave interactions, including collision scenarios and parameter-induced transitions (e.g., Akhmediev breather cases). Overall, the work broadens the catalog of exact, multi-component, bounded solutions for vector cmKdV systems and supplies a versatile algebraic toolkit for studying complex nonlinear wave interactions.

Abstract

This paper constructs the $N$-fold Darboux transformation (DT) for the vector complex modified Korteweg-de Vries (vcmKdV) equation and presents its determinant representation. Utilizing the DT and multi-fold eigenvalue degeneracy, we derive globally bounded solutions for the vcmKdV equation, including $N$-bright-bright-bright solitons, $N$-dark-bright-bright solitons, $N$-breathers, $N$-positon solutions, and $N$th-order rogue wave solutions." All these solutions are globally bounded. Graphical representations of bright-bright-bright and dark-bright-bright soliton solutions are provided, illustrating phenomena where periodic oscillatory waves coexist or interact with solitons. The collision scenarios of the two-bright-bright-bright solution have been investigated by using the asymptotic analysis. The bounded Akhmediev breather, the bounded breather with dark-bright soliton and breather-breather mixed waves are graphically shown. We give the graphs of the positon solution, the rogue wave and the rogue wave mixes with dark-bright solitons and breathers.

Figures (13)

• Figure 1: The bounded one-bright-bright-bright solution of vcmKdV with $\zeta_1 = \dfrac{1}{2} + \dfrac{1}{4}i$, $c_1=1, c_2=1, c_3=2$
• Figure 2: Solutions with the choice of $\zeta_1 = -\zeta_2$.Panel (a) is the bounded periodic solution with $\zeta_1 = \dfrac{1}{2} + \dfrac{1}{4}{\rm i}$, $\zeta_2 = -\dfrac{1}{2} - \dfrac{1}{4}{\rm i}$, $c_1 = 1$, $c_2 = 1$, $c_3 = 3$. Panel (b) is the bounded two bright solitons with a periodic solution with $\zeta_1 = \dfrac{1}{2} + \dfrac{1}{4}{\rm i}$, $\zeta_2 = -\dfrac{1}{2} - \dfrac{1}{4}{\rm i}$, $c_1 = 1$, $c_2 = 1$, $c_3 = 50$.
• Figure 3: Panel (a), (b), (c) are the 3D plots of the bounded two-bright-bright-bright solution of vcmKdV with $\zeta_1 = \dfrac{4}{7} + \dfrac{4}{7}i$, $\zeta_2 = -\dfrac{4}{7} - \dfrac{8}{7}i$, $c_1 = 1$, $c_2 = \dfrac{1}{10}$, $c_3 = \dfrac{1}{100}$. (d), (e), (f) are the profiles of the bounded two-bright-bright-bright solution at $t=0, t=-2, t=2$.
• Figure 4: The bounded one-dark-bright-bright solution of vcmKdV with $\zeta_1 = \dfrac{1}{2} + \dfrac{1}{4}\mathrm{i}$, $c_1=1, c_2=1, c_3=2$
• Figure 5: Panel (a), (b), (c) are the 3D plots of the bounded two-dark-bright-bright solution of vcmKdV with $\zeta_1 = \dfrac{4}{9} + \dfrac{23}{27}{\rm i}$, $\zeta_2 = -\dfrac{4}{9} - \dfrac{23}{54}{\rm i}$,$c_1 = \dfrac{1}{10}$, $c_2 = \dfrac{1}{10}$, $a= \dfrac{1}{10}$. Panel (d), (e), (f) are the profiles of the bounded two-bright-bright-bright solution at $t=0, t=-6, t=4$.

Theorems & Definitions (3)

• Proposition 2.1
• Proposition 2.2
• proof