Computing the Dynamics of Multi-Lumps in Nonlinearity-Managed Spatial-Symmetric Dispersive Wave Framework

TL;DR

This work addresses the dynamics of multi-lump localized waves in a generalized $(3+1)$-D spatially symmetric nonlinear dispersive water-wave framework (GSSNDW). It develops a Hirota bilinear approach combined with generalized polynomial expansions to construct explicit first-, second-, and third-order lump solutions, with dimensional reduction to a $(2+1)$-D KP-type setting. The lumps are shown to be non-interacting and to form geometric patterns (dot, triangular, pentagonal) whose features are tunable by model parameters, and the method extends to arbitrary order. The results offer a systematic pathway to analyzing higher-dimensional lump dynamics in nonlinear dispersive systems and hint at applications to other localized wave phenomena in complex media.

Abstract

We investigate the dynamics of multi-lump waves in a new version of a generalized spatial-symmetric higher-dimensional nonlinear dispersive water wave model using an analytical approach. This involves the proposition of a new spatial-symmetric nonlinear model in (3+1)-dimensions and the construction of its explicit solutions for multi-lump waves through a systematic analytical framework by employing Hirota's bilinear method and generalized polynomial expansions. Analyzing the resultant explicit solutions in terms of their dynamical characteristics reveals that the obtained multi-lump waves are non-interacting and exhibit different geometrical patterns. The observed results demonstrate the significance of new higher-dimensional nonlinear dispersive models in enhancing our understanding of the dynamics of various types of localized waves.

Figures (3)

• Figure 1: Evolution of first-order lump structure in the $x$--$y$ plane at $z = 0.1$ and other remaining parameters are taken as $\alpha_1 = 3$, $\alpha_3 = -2.5$, $\beta_1 = -2$, $\beta_3 = -1$, $\gamma_1 = 3$, $\gamma_3 = -4$ and $c=0.5$. A similar profile evolution can be obtained along $x$--$z$ and $y$--$z$ planes at a fixed $y$ and $x$, respectively.
• Figure 2: The evolution of second-order lump-wave possessing triangular structure in the $x$--$y$ planes for $z = 0.1$ with other parameters taken as $\alpha_{1} = 5,\; \alpha_{3} = -2.5,\; \beta_{3} = -4.5,\; \gamma_{3} = -4,\; c = 0.5,\; \phi = -3,\; \psi = 5,\;$ and $b_{2,0}=c_{2,0} = 100.$
• Figure 3: The evolution of third-order lump-wave solution exhibiting a pentagon structure in the $x$--$y$ planes for $z = 0.1$ with other parameters as $\alpha_{1} = 0.2,\; \alpha_{3} = -0.1,\; \beta_{3} = -1.5,\; \gamma_{3} = -0.15,\; c = 0.5,\;$ and $\phi = 200,\; \psi = 100.$