<p>The interaction of nonlinear waves in an inhomogeneous medium generally leads to temporal evolution in velocity, shape, or amplitude. Therefore, the study of solitary waves is prominent for re-emerging feature to its permanent structure after interaction. This work addresses the propagation dynamics of solitary waves for the NNV equation. It is a highly nonlinear two dimensional isotropic Lax integrable generalization of the well-known KdV equation. The symmetry reductions under infinitesimals and the generalized exact solutions of the NNV equation have been well investigated using the Lie symmetry method. The infinitesimals under a one-parameter group of transformations preserve invariance of the test equation and yield symmetry reductions. The twice implementation of symmetry reductions results in a system of ODEs. This system of ODEs is integrated under parametric constraints and provides exact solutions. These solutions are entirely new and more general than previously reported results, owing to arbitrary functions <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$f_1(t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>f</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$f_2(t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>f</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$f_3(t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>f</mi> <mn>3</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, as well as several constants. The generalness of derived results are ensured with deductions of previous results [<CitationRef CitationID="CR8">8</CitationRef>, <CitationRef AdditionalCitationIDS="CR11" CitationID="CR10">10</CitationRef>–<CitationRef CitationID="CR12">12</CitationRef>]. To understand the propagation properties of wave phenomena, these solutions are extended to include a graphical representation, which reveals a novel class of solitons and lump solutions. Moreover, adjoint equations and conserved vectors are derived under the Lagrangian formulation using Ibragimov’s method for each generator.</p>

On Lie symmetries, invariant solutions, conservation laws and soliton dynamics of Nizhnik–Novikov–Veselov equation in oceanic shallow-water

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发表日期：

2026年7月16日

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Dig Vijay Tanwar

摘要

The interaction of nonlinear waves in an inhomogeneous medium generally leads to temporal evolution in velocity, shape, or amplitude. Therefore, the study of solitary waves is prominent for re-emerging feature to its permanent structure after interaction. This work addresses the propagation dynamics of solitary waves for the NNV equation. It is a highly nonlinear two dimensional isotropic Lax integrable generalization of the well-known KdV equation. The symmetry reductions under infinitesimals and the generalized exact solutions of the NNV equation have been well investigated using the Lie symmetry method. The infinitesimals under a one-parameter group of transformations preserve invariance of the test equation and yield symmetry reductions. The twice implementation of symmetry reductions results in a system of ODEs. This system of ODEs is integrated under parametric constraints and provides exact solutions. These solutions are entirely new and more general than previously reported results, owing to arbitrary functions $f_1(t)$ $f_{1} (t)$ , $f_2(t)$ $f_{2} (t)$ , and $f_3(t)$ $f_{3} (t)$ , as well as several constants. The generalness of derived results are ensured with deductions of previous results [8, 10–12]. To understand the propagation properties of wave phenomena, these solutions are extended to include a graphical representation, which reveals a novel class of solitons and lump solutions. Moreover, adjoint equations and conserved vectors are derived under the Lagrangian formulation using Ibragimov’s method for each generator.