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A Modified Davey-Stewartson System of Nonlinear Dust Acoustic Waves in (3+1)-Dimensions: Lie Symmetries and Exact Solutions

Seyma Gonul, Yasin Hasanoglu, Ayse Tiryakioglu, Yasemin Calis, Cihangir Ozemir

TL;DR

This work analyzes a 3+1D modified Davey–Stewartson system derived for nonlinear dust acoustic waves, noting an extra complex-potential term. It employs Lie-group analysis to reveal an infinite-dimensional symmetry algebra, identifying a transform that removes the extra term under a parameter constraint and linking the system to a standard DS framework via a semi-direct product with a Kac–Moody algebra. By applying traveling-wave reductions, the authors construct exact solutions including line solitons and kink–soliton configurations and perform a stability analysis of these traveling waves. The results provide deep symmetry insights for higher-dimensional DS-type PDEs and yield explicit, physically meaningful wave structures relevant to dusty plasma modeling.

Abstract

This article is devoted to the analysis of a modified Davey-Stewartson system in three space dimensions, which was obtained in plasma physics for propagation of nonlinear dust acoustic waves. The system differs from the Davey-Stewartson systems available in the literature by an additional term which can be viewed as a constant complex potential. We show that, under a certain condition on the parameters of the system, this term can be removed by a transformation. This restriction also separates the different realizations of Lie symmetry algebra of the modified Davey-Stewartson system, which is identified as semi-direct sum of a finite-dimensional algebra with a Kac-Moody algebra. Having shed light on the group-theoretical properties of the system, we present several results on the exact solutions of generalized traveling wave type, some of which are line solitons and kink solitons on planes in space. We finalize by analysing the stability of traveling wave solutions.

A Modified Davey-Stewartson System of Nonlinear Dust Acoustic Waves in (3+1)-Dimensions: Lie Symmetries and Exact Solutions

TL;DR

This work analyzes a 3+1D modified Davey–Stewartson system derived for nonlinear dust acoustic waves, noting an extra complex-potential term. It employs Lie-group analysis to reveal an infinite-dimensional symmetry algebra, identifying a transform that removes the extra term under a parameter constraint and linking the system to a standard DS framework via a semi-direct product with a Kac–Moody algebra. By applying traveling-wave reductions, the authors construct exact solutions including line solitons and kink–soliton configurations and perform a stability analysis of these traveling waves. The results provide deep symmetry insights for higher-dimensional DS-type PDEs and yield explicit, physically meaningful wave structures relevant to dusty plasma modeling.

Abstract

This article is devoted to the analysis of a modified Davey-Stewartson system in three space dimensions, which was obtained in plasma physics for propagation of nonlinear dust acoustic waves. The system differs from the Davey-Stewartson systems available in the literature by an additional term which can be viewed as a constant complex potential. We show that, under a certain condition on the parameters of the system, this term can be removed by a transformation. This restriction also separates the different realizations of Lie symmetry algebra of the modified Davey-Stewartson system, which is identified as semi-direct sum of a finite-dimensional algebra with a Kac-Moody algebra. Having shed light on the group-theoretical properties of the system, we present several results on the exact solutions of generalized traveling wave type, some of which are line solitons and kink solitons on planes in space. We finalize by analysing the stability of traveling wave solutions.
Paper Structure (8 sections, 3 theorems, 62 equations, 3 figures)

This paper contains 8 sections, 3 theorems, 62 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Group-Theoretical Analysis
  3. Transformation
  4. Lie symmetry algebra
  5. Exact Solutions
  6. Traveling wave solutions
  7. Stability Analysis
  8. Conclusion

Key Result

Proposition 2.1

When $a_3\neq 0$, if $b_1+2a_1=0$, the transformation from 31MDS to 31MDS2 is achieved through where

Figures (3)

  • Figure 1: Solution \ref{['sechn']} on the plane $z=0$; (a) $t=0$, (b) $t=0$, (c) $t=5$, (d) $t=10$.
  • Figure 2: Solution \ref{['sech2n']} on the plane $z=0$; (a) $t=0$, (b) $t=0$, (c) $t=5$, (d) $t=10$.
  • Figure 3: (a) $A=1$, $L=4$ (b) $A=-1$, $L=-4$, (c) and (e) $A=1$, $L=-4$, (d) and (f) $A=-1$, $L=4$.

Theorems & Definitions (4)

  • Proposition 2.1
  • Theorem 2.1
  • Corollary 2.1
  • Remark 2.1