Localized excitation on the Jacobi elliptic periodic background for the (n+1)-dimensional generalized Kadomtsev-Petviashvili equation

TL;DR

This work addresses nonlinear wave dynamics on a Jacobi elliptic periodic background for the $$(n+1)$$-dimensional gKP equation by combining a Lax-pair framework with Lamé-equation spectral analysis and Darboux transformations to produce explicit breather solutions. A Jacobi-elliptic seed is evolved through one- and two-fold Darboux transformations to generate first- and second-order bright and dark breathers, with degenerate limits recovering solitons. The study reveals that the nature and velocity of breathers are governed by the spectral parameter regimes and a dispersion parameter $D=-\sum_{i=2}^{n} \sigma_i \omega_i$, illustrating coupling between longitudinal and transverse dispersive channels. These results advance exact solutions on nonconstant elliptic backgrounds for high-dimensional KP-type systems, with potential implications for fluid mechanics and related nonlinear wave phenomena.

Abstract

In this paper, the linear spectral problem, which associated with the (n+1)-dimensional generalized Kadomtsev-Petviashvili (gKP) equation, with the Jacobi elliptic function as the external potential is investigated based on the Lamé function, from which some novel local nonlinear wave solutions on the Jacobi elliptic function have been obtained by Darboux transformation, and the corresponding dynamics have also been discussed. The degenerate solutions of the nonlinear wave solutions on the Jacobi function background for the gKP equation are constructed by taking the modulus of the Jacobi function to be 0 and 1. The findings indicate that there can be various types of nonlinear wave solutions with different ranges of spectral parameters, including soliton and breather waves. Furthermore, the interplay between nonlinearity and dispersion is found to have observable effects on the propagation dynamics of breather waves. These results will be useful for elucidating and predicting nonlinear phenomena in related physical fields, such as fluid mechanics and physical ocean.

Figures (6)

• Figure 1: Propagation profiles and corresponding contour plots of the breather wave solution \ref{['dt1snu1']}. The parameters are chosen as $k=0.3$, $\alpha=\pi/3$, $\beta=1$, $\lambda=1.8$, $\delta=1$, $\omega_2=\omega_3=1$, and $D=5$. (a) Evolution in the $x_1$--$t$ plane with $x_2=x_3=1$; (b) evolution in the $x_2$--$t$ plane with $x_1=x_3=1$; (c) evolution in the $x_3$--$t$ plane with $x_1=x_2=1$.
• Figure 2: Propagation profiles and corresponding contour plots of the dark breather solution \ref{['dt1snu1d']}. The parameters are chosen as $k=0.6$, $\alpha=\pi/3$, $\beta=1$, $\lambda=-2.1$, $\delta=1$, $\omega_2=\omega_3=1$, and $D=5$. (a) Evolution in the $x_1$--$t$ plane with $x_2=x_3=1$; (b) evolution in the $x_2$--$t$ plane with $x_1=x_3=1$; (c) evolution in the $x_3$--$t$ plane with $x_1=x_2=1$.
• Figure 3: Propagation profiles and corresponding contour plots of the nonlinear wave solution \ref{['dt2snu2']} generated via the two-times Darboux transformation. The parameters are chosen as $k=0.3$, $\alpha=\pi/3$, $\beta=1$, $\lambda_1=1.8$, $\lambda_2=1.4$, $\delta=1$, $\omega_2=\omega_3=1$, and $D=5$. (a) Evolution in the $x_1$--$t$ plane with $x_2=x_3=1$; (b) evolution in the $x_2$--$t$ plane with $x_1=x_3=1$; (c) evolution in the $x_3$--$t$ plane with $x_1=x_2=1$.
• Figure 4: Propagation profiles and corresponding contour plots of the nonlinear wave solution \ref{['dt2snu2d']} generated via the two-times Darboux transformation. The parameters are chosen as $k=0.6$, $\alpha=\pi/3$, $\beta=1$, $\lambda_1=-2.1$, $\lambda_2=-2.3$, $\delta=1$, $\omega_2=\omega_3=1$, and $D=5$. (a) Evolution in the $x_1$--$t$ plane with $x_2=x_3=1$; (b) evolution in the $x_2$--$t$ plane with $x_1=x_3=1$; (c) evolution in the $x_3$--$t$ plane with $x_1=x_2=1$.
• Figure 5: Propagation and density plots for the degenerate solutions of nonlinear wave solutions \ref{['dt1snu1']} and \ref{['dt2snu2']} based on $k=0$, (a) Ploting from solution \ref{['degedt1k0']}; (b) Ploting from solution \ref{['degedt2k0']}.

Theorems & Definitions (13)

• Theorem 1
• Theorem 2
• proof
• Proposition 1
• proof
• Proposition 2
• Theorem 3
• Proposition 3
• proof
• Theorem 4