Nizhnik Novikov Veselov Equation Research Articles

Painlevé analysis, Gram determinant solution and the dynamic behavior of the (2+1)-dimensional variable-coefficient asymmetric Nizhnik–Novikov–Veselov equation

Painlevé analysis, Gram determinant solution and the dynamic behavior of the (2+1)-dimensional variable-coefficient asymmetric Nizhnik–Novikov–Veselov equation

New patterns of localized excitations in (2+1)-dimensions: The fifth-order asymmetric Nizhnik–Novikov–Veselov equation

By applying the mastersymmetry of degree one to the time-independent symmetry K 1, the fifth-order asymmetric Nizhnik–Novikov–Veselov system is derived. The variable separation solution is obtained by using the truncated Painlevé expansion with a special seed solution. New patterns of localized excitations, such as dromioff, instanton moving on a curved line, and tempo-spatial breather, are constructed. Additionally, fission or fusion solitary wave solutions are presented, graphically illustrated by several interesting examples.

Application of Bell polynomial in the generalized (2+1)-dimensional Nizhnik–Novikov–Veselov equation

Application of Bell polynomial in the generalized (2+1)-dimensional Nizhnik–Novikov–Veselov equation

Precise invariant travelling wave soliton solutions of the Nizhnik–Novikov–Veselov equation with dynamic assessment

The (2 + 1)-dimensional Nizhnik–Novikov–Veselov equation is a significant model that describes the behaviour of conserved scalar nucleons and their connection to neutral scalar masons. This system exhibits self-formation and depends on various free parameters. It is commonly used to study the dynamics of scalar nucleons and neutral scalar masons. This research article used an extended direct algebraic technique to obtain exact travelling wave solutions for the Nizhnik–Novikov–Veselov system. The obtained soliton solutions include various types, such as multi-periodic, periodic, plane, singular, bright, dark, and flat kink-type wave solitons. We present these soliton solutions graphically by varying the involved parameters using the advanced software program Wolfram Mathematica. The graphical representations in 3D, contour, and 2D surfaces allow us to visualise the behaviour of the solutions as the parameters change and the physical interpretation for these solutions is obtained. Additionally, we conduct a sensitivity analysis to examine the wave profiles of the newly designed dynamical framework. The results of this analysis demonstrate the reliability and efficiency of the proposed method, which can be applied to find closed-form travelling wave solitary solutions for a wide range of nonlinear evolution equations.

Non classical interaction aspects to a nonlinear physical model

This study deals the dynamics of waves to the conformable fractional (2+1)-dimensional Nizhnik–Novikov–Veselov (NNV) equations. The (2+1)-dimensional NNV equations are the isotropic Lax integrable extension of the (1+1)-dimensional Korteweg–de Vries equations. Fractional differential models (FDMs) from the corresponding integer order model can describes more complex behavior and even cover all properties of integer order model. By the usage of different test approaches, the Hirota bilinear method (HBM) has been successfully applied. The Hirota method is a well-known and reliable mathematical tool for finding the soliton solutions of fractional nonlinear partial differential equations in many fields. However, it demands bilinearization of nonlinear fractional PDE. A class of results in the shapes of lump-periodic, breather-type and two wave solutions have been extracted. Numerical visualizations of the results are also used to demonstrate the implications of the fractionality and parameters. It is acceptable to assume that many experts in engineering models will learn from this research. The results prove the used algorithm is effective, quick, easy, and flexible in its application to various systems.

Investigation of solitary wave structures for the stochastic Nizhnik–Novikov–Veselov (SNNV) system

The current study deals with the stochastic Nizhnik–Novikov–Veselov (SNNV) system analytically under the multiplicative noise effect. The Nizhnik–Novikov–Veselov equation is an extension of the KdV equation with applications in shallow-water waves, ionic acoustic waves in plasma, long internal waves in density-stratified oceans, and sound waves on crystal networks. The stochastic wave structures are constructed with the help of the Sardar subequation method. The different solutions are extracted which are in the form of solitons and solitary wave structures in the noise effect. Additionally, the stochastic behavior appears in the 3-dim, 2-dim, and their corresponding contour representations by the various selection of parameters.

Dynamical behaviors with various exact solutions to a (2+1)-dimensional asymmetric Nizhnik–Novikov–Veselov equation using two efficient integral approaches

The main goal of this research work is to explore novel soliton solutions and dynamics of solitonic structures to asymmetric Nizhnik–Novikov–Veselov (ANNV) equations by two methods, namely, the generalized exponential rational function (GERF) method and the modified extended tanh expansion (METE) method. These techniques are the most effective and reliable tools for solving nonlinear equations with partial derivatives. The used methods provide a wide range of soliton solutions that can motivate applied scientists and researchers for these specific structures. These acquired solutions are in the form of exponential, trigonometric and hyperbolic functions including, [Formula: see text] [Formula: see text] and of their combinations. It has also been observed that the obtained soliton solutions of the ANNV equations are bell-shape, anti-bell-shape, periodic solution, multisoliton and different types of soliton solutions. To illustrate the physical features and dynamics characteristics of some obtained solutions, three-dimensional (3D) and two-dimensional (2D) figures are exhibited through the Mathematica software. The methods applied in this paper are suitable to study the soliton solutions for the ANNV equation without producing the complexities in some other known analytical method. Finally, the employed methods can be earmarked easily for solving various classes of nonlinear evolution equations appearing in mathematical physics, fluid dynamics, plasma physics, nonlinear waves and nonlinear sciences.

Identifying Combination of Dark–Bright Binary–Soliton and Binary–Periodic Waves for a New Two-Mode Model Derived from the (2 + 1)-Dimensional Nizhnik–Novikov–Veselov Equation

In this paper, we construct a new two-mode model derived from the (2+1)-dimensional Nizhnik–Novikov–Veselov (TMNNV) equation. We generalize the concept of Korsunsky to accommodate the derivation of higher-dimensional two-mode equations. Since the TMNNV is presented here, for the first time, we identify some of its solutions by means of two recent and effective schemes. As a result, the Kudryashov-expansion method exports a combination of bright–dark binary solitons, which simulate many applications in optics, photons, and plasma. The modified rational sine and cosine functions export binary–periodic waves that arise in the field of surface water waves. Moreover, by using 2D and 3D graphs, some physical properties of the TMNNV were investigated by means of studying the effect of the following parameters of the model: nonlinearity, dispersion, and phase–velocity. Finally, we checked the validity of the obtained solutions by verifying the correctness of the original governing model.

New Complex Wave Solutions and Diverse Wave Structures of the (2+1)-Dimensional Asymmetric Nizhnik–Novikov–Veselov Equation

In this paper, we use a new, extended Jacobian elliptic function expansion method to explore the exact solutions of the (2+1)-dimensional asymmetric Nizhnik–Novikov–Veselov (aNNV) equation, which is a nonlinear physical model to describe an incompressible fluid. Combined with the mapping method, many new types of exact Jacobian elliptic function solutions are obtained. As we use two new forms of transformation, most of the solutions obtained are not found in previous studies. To show the complex nonlinear wave phenomena, we also provide various wave structures of a group of solutions, including periodic wave and solitary wave structures of ordinary traveling wave solutions, horseshoe-type wave, s-type wave and breaker-wave structures superposed by two kinds of waves: chaotic wave structures with chaotic behavior and spiral wave structures. The results show that this method is effective and powerful and can be used to construct various exact solutions for a wide range of nonlinear models and complex nonlinear wave phenomena in mathematical and physical research.

Interaction solutions and molecule state between resonance Y-type solitons and lump waves, and transformed 2-breather molecular waves of a (2+1)-dimensional asymmetrical Nizhnik–Novikov–Veselov equation

Interaction solutions and molecule state between resonance Y-type solitons and lump waves, and transformed 2-breather molecular waves of a (2+1)-dimensional asymmetrical Nizhnik–Novikov–Veselov equation

Various dynamic behaviors to the (2+1)-dimensional Nizhnik–Novikov–Veselov equations in incompressible fluids

In this paper, under investigation is the (2+1)-dimensional Nizhnik–Novikov–Veselov equations. The breather wave solutions are derived via the three-wave method. The double-periodic soliton solutions are studied based on a direct function method. Furthermore, the interaction solutions of the lump and periodic waves are investigated by a special function consist of the quadratic polynomial functions and periodic functions. At the same time, we also choose some specific examples to illustrate the dynamic behaviors of nonlinear waves.

Dynamics of mixed lump-soliton for an extended (2+1)-dimensional asymmetric Nizhnik–Novikov–Veselov equation

A new (2+1)-dimensional higher-order extended asymmetric Nizhnik–Novikov–Veselov (eANNV) equation is proposed by introducing the additional bilinear terms to the usual ANNV equation. Based on the independent transformation, the bilinear form of the eANNV equation is constructed. The lump wave is guaranteed by introducing a positive constant term in the quadratic function. Meanwhile, different class solutions of the eANNV equation are obtained by mixing the quadratic function with the exponential functions. For the interaction between the lump wave and one-soliton, the energy of the lump wave and one-soliton can transfer to each other at different times. The interaction between a lump and two-soliton can be obtained only by eliminating the sixth-order bilinear term. The dynamics of these solutions are illustrated by selecting the specific parameters in three-dimensional, contour and density plots.

Nonlinear superposition between lump waves and other waves of the (2 $$+$$ 1)-dimensional asymmetrical Nizhnik–Novikov–Veselov equation

Nonlinear superposition between lump waves and other waves of the (2 $$+$$ 1)-dimensional asymmetrical Nizhnik–Novikov–Veselov equation

D’Alembert wave and soliton molecule of the generalized Nizhnik–Novikov–Veselov equation

Traveling wave solution is one of the effective methods for solving nonlinear partial differential equations. D’Alembert solution is a special kind of traveling wave solution. There have been many studies about D’Alembert solution. In this paper, we will solve D’Alembert-type wave solutions for (2+1)-dimensional generalized Nizhnik–Novikov–Veselov equation. Based on the Hirota bilinear transformation and velocity resonance mechanism, the states of soliton molecules composed of two solitons, three solitons and four solitons are studied. It is concluded that D’Alembert-type wave is closely related to soliton molecules.

ON THE COMPLEX MIXED DARK-BRIGHT WAVE DISTRIBUTIONS TO SOME CONFORMABLE NONLINEAR INTEGRABLE MODELS

In this research paper, we implement the sine-Gordon expansion method to two governing models which are the (2+1)-dimensional Nizhnik–Novikov–Veselov equation and the Caudrey–Dodd–Gibbon–Sawada–Kotera equation. We use conformable derivative to transform these nonlinear partial differential models to ordinary differential equations. We find some wave solutions having trigonometric function, hyperbolic function. Under the strain conditions of these solutions obtained in this paper, various simulations are plotted.

Resonance [formula omitted]-type soliton and hybrid solutions of a (2+1)-dimensional asymmetrical Nizhnik–Novikov–Veselov equation

In this paper, a (2+1)-dimensional asymmetrical Nizhnik–Novikov–Veselov equation is studied. By adding some new constraints to the N-soliton solutions, the resonance Y-type soliton and the interaction solutions between M-resonance Y-type solitons and P-resonance Y-type solitons are obtained. The new types of solutions including the double openings of resonance Y-type soliton solutions and mixed solutions of resonance solitons and breathers are presented. The hybrid solutions consisting of the resonance Y-type solitons and the multiple lump waves are constructed by using restrictions and the long wave limit method. The trajectories of lump waves in hybrid waves are analyzed from the point of view of mathematical mechanism. The results are helpful to explain some new nonlinear phenomena in the theory of solitons.

Soliton molecules in the (2+1)-dimensional Nizhnik–Novikov–Veselov equation

The [Formula: see text]-soliton solution of the (2+1)-dimensional Nizhnik–Novikov–Veselov equation is constructed. The line soliton molecule, the breather and the lump soliton are presented successively for [Formula: see text]. The three-soliton molecule structure, interaction of one-soliton molecule and a line soliton, the soliton molecules consisting of a line soliton and the breather/lump soliton of the solution [Formula: see text] are constructed for [Formula: see text]. Moreover, the four-soliton molecule structure, interaction of the soliton molecule and a line soliton, the soliton molecule consisting of the line soliton molecule and a lump soliton, the elastic interaction between the line soliton molecule and a lump soliton, the soliton molecules consisting of the line soliton molecule and the breather, two breather solitons, the breather soliton and a lump of the variable [Formula: see text] for this equation are also derived for [Formula: see text] by applying the velocity resonance, the module resonance of wave number and the long-wave limit ideas. To illustrate these phenomena, the analysis explicit solutions are all given and their dynamics features are all displayed through figures.

The lump solutions for the (2+1)-dimensional Nizhnik–Novikov–Veselov equations

The exact solutions for the (2+1)-dimensional Nizhnik–Novikov–Veselov (NNV) equations are investigated via Maple software. According to the Hirota bilinear form of the NNV equations, lump solutions are mainly obtained by the limit method of the heteroclinic test functions with initial values. Furthermore, the interaction solutions of the lump and N-soliton solutions are researched with the aid of new test functions which consist of the quadratic polynomial functions and the exponential functions. Especially, the sufficient conditions for the existence of the interactions solutions for NNV equations are given.

Numerical soliton solutions of fractional (2 + 1)-dimensional Nizhnik–Novikov–Veselov equations in nonlinear optics

In this paper, time-fractional (2 + 1)-dimensional Nizhnik–Novikov–Veselov equations have been solved numerically utilizing the Kansa method, in which the multiquadrics are taken as radial basis function. To attain this, a numerical scheme based on finite difference and Kansa method has been proposed. In addition, the soliton solutions have been obtained by employing Kudryashov method and tanh method for comparison purpose with the obtained numerical solutions. The numerical examples are given to demonstrate the accuracy and applicability of the proposed method.

D’Alembert wave and soliton molecule of the modified Nizhnik–Novikov–Veselov equation

The wave motion equation is one of the fundamental equations to describe vibrations of continuous systems. The D’Alembert solution of the wave motion equation is an important basic formula in linear partial differential equations. The study of the D’Alembert wave deserves deep consideration in nonlinear equations. In this paper, the D’Alembert-type wave of the (2 + 1)-dimensional modified Nizhnik–Novikov–Veselov (mNNV) equation is derived by the Ansatze method. The Hirota bilinear form of the mNNV equation is constructed by introducing the dependent variable transformation. The multi-soliton solution is obtained by solving the corresponding bilinear form. By combining the velocity resonance mechanism, a three-soliton molecule, the interaction between a soliton molecule and one soliton, and the interaction between a soliton–solitoff molecule and one soliton of the mNNV equation are obtained. The dynamics of these solutions are shown by selecting the appropriate parameters. These phenomena for the mNNV equation have not yet been given via other methods.