Lie symmetry transformation, conservation laws and nonlinear self-adjointness of (2+1)-dimensional Date–Jimbo–Kashiwara–Miwa equation

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The objective of the present article is to seek new explicit solutions to the (2+1)-dimensional Nizhnik-Novikov-Veselov(NNV) equation. The NNV system is highly nonlinear in nature and is a known isotropic Lax extension of the Korteweg-de Vries model. The similarity transformation method has been used to systematically reduce the NNV equation into ordinary differential equations(ODEs). The new exact solutions have been obtained by solving the obtained ODEs based on the formed relationships. The resulting soliton solutions contain some arbitrary constants and functions. The use of appropriate functions and constants highlighted that the solutions of the NNV equation might be soliton, multisoliton, parabolic, doubly soliton and trigonometric. Moreover, the stability of the corresponding dynamical system has been investigated using bifurcation theory with different parametric regions. The figures obtained during MATLAB simulation supported the dynamic features of the derived solutions. Finally, we explore incredible aspects of the exact wave solutions via phase portraits. The phase portraits validate the existence of some families of homoclinic and periodic orbits about the equilibrium points, respectively.

Section: Existence and Stability Of Equilibriamentioning

confidence: 99%

Soliton solutions of the (2+1)-dimensional Nizhnik-Novikov-Veselov equation via the Lie symmetry method and its stability analysis by using bifurcation theory

Manju¹,

Kumar²

2022

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“…Also, some third order difference equations were studied using that technique. Several investigations on differential equations can be pursued in accordance with the studies mentioned [20][21][22].…”

Section: Introductionmentioning

confidence: 99%

First integrals, conserved vectors of nonlinear partial difference equations

Hussain,

Kara,

Zaman

2024

We perform a symmetry analysis of some nonlinear partial difference equations (nP$\triangle$Es), where the discrete version is obtained using some discretization approach. The discrete versions of the wave, diffusion, Fisher and Huxley equations are the subject of this research. At first, the initial invariance approach is the Lie symmetry approach. The first integrals technique that Hydon introduced to be used with discrete ordinary difference equations (O$\triangle$Es) serves as our inspiration in this situation. We develop a similar technique for generating the first integral vectors of the nP$\triangle$Es without recourse to symmetry generators.

“…These invariant solutions describe their behavior like they display its asymptotic nature and the information of the singularities if it exists. Thus, various efficient methods and techniques are there to obtain exact soliton solutions, such as inverse scattering transformation [1], Hirota bilinear method [2], nonlocal symmetry [3], Bäcklund transformation [3], Painlevé analysis [3], Lie symmetry method [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22], etc.…”

Section: Introductionmentioning

confidence: 99%

“…Therefore, the group invariance soliton solutions can be derived. The description of the procedure and its utilization one can easily see in the pieces of literature [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22].…”

Section: Introductionmentioning

confidence: 99%

Symmetry analysis, optimal classification and dynamical structure of exact soliton solutions of (2+1)-dimensional modified Bogoyavlenskii–Schiff equation

Kumar

Manju

2022

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The present research framework looks over complete sorted symmetry group classification and optimal subalgebras of (2+1)-dimensional modified Bogoyavlenskii–Schiff(mBSchiff) equation. It’s highly nonlinear and exhibits wave propagation in thermal pulse, sound wave, and bound particle. Using the invariance property of Lie groups, adequate infinitesimal symmetry of Lie algebra has been set up for the mBSchiff equation. A rigorous and systematized algorithm is carried out to obtain one optimal system based on the invariance feature of adjoint transformation. Further, symmetry reduction of the mBSchiff equation has been made to derive a system of ordinary differential equations with newly established similarity variables. The complete set of group invariant solutions for each corresponding subalgebras has been made. The derived solutions have diverse physical phenomena, which MATLAB simulation can quickly analyze. Thus, solutions presented here are kink, positon, soliton, doubly soliton, negaton, multisoliton types, which add on some meaningful physical aspects of the research.