Kink breathers on a traveling wave background in the defocusing modified Korteweg--de Vries equation

TL;DR

The paper addresses constructing kink breathers on traveling periodic backgrounds for the defocusing mKdV equation by representing general traveling waves with genus-two Riemann theta data via quotients of Jacobi theta functions. It develops a two-parameter representation of the traveling wave, derives explicit eigenfunctions at the origin of the Lax pair, and uses a Darboux transform to produce a bounded kink breather on the elliptic background, with the breather speed and localization given in closed form. The results unify multiple elliptic-function representations (root-based, Weierstrass, and Jacobi theta) and show that the kink breather reduces to a two-soliton solution in the hyperbolic limit $k\to1$, thereby connecting to known NLS/mKdV breathers. The work solves the open problem posed in MP24 on kink breathers for the defocusing mKdV by providing explicit genus-two theta expressions and a rigorous construction framework, with implications for numerical and lab experiments in dispersive hydrodynamics.

Abstract

We characterize a general traveling periodic wave of the defocusing mKdV (modified Korteweg--de Vries) equation by using a quotient of products of Jacobi's elliptic theta functions. Compared to the standing periodic wave of the defocusing NLS (nonlinear Schrödinger) equation, these solutions are special cases of Riemann's theta function of genus two. Based on our characterization, we derive a new two-parameter solution form which defines a general three-parameter solution form with the scaling transformation. Eigenfunctions of the Lax system for the general traveling periodic wave are also characterized as quotients of products of Jacobi's theta functions. As the main outcome of our analytical computations, we derive a new solution of the defocusing mKdV equation which describes the kink breather propagating on a general traveling wave background.

Figures (7)

• Figure 1: Domain $\Omega$ (green shaded area) between the two boundaries (blue solid lines). Example \ref{['ex-snoidal']} corresponds to the red dotted line.
• Figure 2: Profiles of $\phi$ versus $x$ for $\zeta_1 = 2$, $\zeta_2 = 1$, $\zeta_3 = 0.5$ (left) and $\zeta_1 = 1$, $\zeta_2 = 0.75$, $\zeta_3 = 0.5$ (right).
• Figure 3: Lax spectrum for $\zeta_1 = 2$, $\zeta_2 = 1$, and $\zeta_3 = 0$ (left) and for $\zeta_1 = 2$, $\zeta_2 = 1$, and $\zeta_3 = 0.5$ (right). Red dots show location of $\pm \zeta_1$, $\pm \zeta_2$, and $\pm \zeta_3$.
• Figure 4: Plots of $u(x,t)$ for the kink breather solution (\ref{['kink-breather']}) with $\zeta_1 = 1$, $k = 0.9$, and $\alpha = 0.25 K$ versus $\xi = x + ct$ for $t = -3$ (left) and $t = 3$ (right). The kink moves to the right relative to the periodic wave and flips its sign. The periodic wave impares the phase shift (\ref{['limits-breather']}) due to the interaction with the kink.
• Figure 5: Plots of $u(x,t)$ for the two-soliton solution (\ref{['two-solitons']}) with $\alpha = 0.5$ versus $\eta = x + 2t$ for $t = -3$ (left), $t = 0$ (middle), and $t = 3$ (right). The soliton with the hyperbolic profile (\ref{['soliton']}) moves to the left relative to the kink and flips the sign after the interaction with the kink. Both the soliton and the kink impares the phase shifts due to the interaction.

Theorems & Definitions (52)

• Remark 2.1
• Example 2.1
• Theorem 1
• Remark 2.2
• Example 2.2
• Corollary 1
• proof
• Example 2.3
• Remark 2.3
• Theorem 2