Nizhnik Novikov Veselov System Research Articles

Localized stem structures in quasi-resonant two-soliton solutions for the asymmetric Nizhnik–Novikov–Veselov system

Elastic collisions of solitons generally have a finite phase shift. When the phase shift has a finitely large value, the two vertices of the (2 + 1)-dimensional two-soliton are significantly separated due to the phase shift, accompanied by the formation of a local structure connecting the two V-shaped solitons. We define this local structure as the stem structure. This study systematically investigates the localized stem structures between two solitons in the (2 + 1)-dimensional asymmetric Nizhnik–Novikov–Veselov system. These stem structures, arising from quasi-resonant collisions between the solitons, exhibit distinct features of spatial locality and temporal invariance. We explore two scenarios: one characterized by weakly quasi-resonant collisions (i.e. a12 ≈ 0), and the other by strongly quasi-resonant collisions (i.e. a12 ≈ +∞). Through mathematical analysis, we extract comprehensive insights into the trajectories, amplitudes, and velocities of the soliton arms. Furthermore, we discuss the characteristics of the stem structures, including their length and extreme points. Our findings shed new light on the interaction between solitons in the (2 + 1)-dimensional asymmetric Nizhnik–Novikov–Veselov system.

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.

Chaotic Phenomena, Sensitivity Analysis, Bifurcation Analysis, and New Abundant Solitary Wave Structures of The Two Nonlinear Dynamical Models in Industrial Optimization

In this research, we discussed the different chaotic phenomena, sensitivity analysis, and bifurcation analysis of the planer dynamical system by considering the Galilean transformation to the Lonngren wave equation (LWE) and the (2 + 1)-dimensional stochastic Nizhnik–Novikov–Veselov System (SNNVS). These two important equations have huge applications in the fields of modern physics, especially in the electric signal in data communication for LWE and the mechanical signal in a tunnel diode for SNNVS. A different chaotic nature with an additional perturbed term was also highlighted. Concerning the theory of the planer dynamical system, the bifurcation analysis incorporating phase portraits of the dynamical systems of the declared equations was performed. Additionally, a sensitivity analysis was used to monitor the sensitivity of the mentioned equations. Also, we extracted new, abundant solitary wave structures with the graphical phenomena of the mentioned nonlinear mathematical models. By conducting an expansion method on the abovementioned equations, we generated three types of soliton structures, which are rational function, trigonometric function, and hyperbolic function. By simulating the 3D, contour, and 2D graphs of these obtained solitons, we scrutinized the behavior of the waves affecting the nonlinear terms. The figures show that the solitary waves obtained from LWE are efficient in analyzing electromagnetic wave signals in the cable lines, and the solitary waves from SNNVS are essential in any stochastic system like a sound wave. Moreover, by taking some values of the parameters, we found some interesting soliton shapes, such as compaction soliton, singular periodic solution, bell-shaped soliton, anti-kink-shaped soliton, one-sided kink-shaped soliton, and some flat kink-shaped solitons, etc. This article will have a great impact on nonlinear science due to the new solitary wave structures with different complex phenomena, sensitivity analysis, and bifurcation analysis.

Exploring stochastic dynamics with different wave structures for the Nizhnik–Novikov–Veselov system and their applications

Exploring stochastic dynamics with different wave structures for the Nizhnik–Novikov–Veselov system and their applications

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.

Solitary wave structures for the stochastic Nizhnik–Novikov–Veselov system via modified generalized rational exponential function method

In this article, a significant stochastic Nizhnik–Novikov–Veselov (SNNV) system with truncated M-fractional derivative (TMD) is investigated. This mathematical model is considered an isotropic Lax extension of the one-dimensional Kortewegde Vries equation which has diverse applications in the areas of warm ions, shallow-water waves, electromagnetic signals, and river irrigation flows. To collect different solitary wave solutions of the SNNV system, a modification of generalized rational exponential function method is employed. Some new trigonometric, exponential and hyperbolic stochastic solutions are obtained by using the modified generalized rational exponential function method (mGERFM). Furthermore, some graphs of the formulated solutions are presented to depict the physical configuration of stochastic solutions by changing the parameters. The method used in this research is effective, precise, competent, and reliable to compute the soliton solutions for nonlinear models. We predict that the performed results will have great potential applications to optical fibers, magneto-electrodynamics, the collision of heavy ions, and quantum mechanics.

Dynamics of the rogue lump in the asymmetric Nizhnik–Novikov–Veselov system

AbstractThe (2 + 1)‐dimensional asymmetric Nizhnik–Novikov–Veselov (ANNV) equation describes an incompressible fluid such as waves with weakly nonlinear restoring forces, long internal waves in a density‐stratified ocean, or acoustic waves on a crystal lattice. In this paper, we construct a new kind of solutions doubly localized in both time and space, that is, a rogue lump wave, of the ANNV equation using the binary Darboux transformation (BDT). This solution is expressed by a class of semirational functions, in which lumps appear from one line soliton and then rapidly disappear into another line soliton. Phase parameter newly introduced in eigenfunctions plays a vital role in controlling the distinct dynamical behaviors of these rogue lumps. The explicit formulas of the approximate locations and heights of rogue waves, lumps, and line solitons are given by asymptotic analysis. Their collision process is discussed in detail. These results further help us to understand the extreme and transient waves in a variety of physical systems.

An investigation of (2+1)-dimensional asymmetric Nizhnik–Novikov–Veselov system: Lie symmetry reductions, invariant solutions, dynamical behaviors and conservation laws

In this study, we develop the asymmetric Nizhnik–Novikov–Veselov (ANNV) system in (2+1)-dimensions, which has applications in processes of interaction of exponentially localized wave structures, as well as the infinitesimal generators, Lie symmetries, vector fields, and the commutator table. The link between Lie symmetry vectors and conserved vectors is constructed using symmetry conservation principles once Lie point symmetries are first deduced. Using the aforementioned Lie symmetry technique, two-stage symmetry reductions are used to obtain the precise analytical answers. These analytical solutions all incorporate a number of different functional parameters as well as arbitrary constant parameters. The diversity of the physical phenomena of the obtained soliton solutions is illustrated by the inclusion of arbitraryness of functional parameters and constants. By using Noether’s method, conservation laws have subsequently been attained. The innovative aspect of the work described in this paper is an attempt to use 3-dimensional and 2-dimensional visuals, along with appropriate arbitrary parameter selections and functional parameter values, to represent the dynamical behavior of the solutions that have been produced. In order to make this research more intriguing, stripe solitons, dark-bright solitons, solitary waves, singular wave-form soliton, and other types of soliton wave profiles of the achieved solutions are described. The effectiveness, benefits, and utility of the employed approach are demonstrated by the physical and graphical interpretation of the answers attained.

The influence of multiplicative noise on the stochastic exact solutions of the Nizhnik–Novikov–Veselov system

In this paper we consider the stochastic Nizhnik–Novikov–Veselov system, which is forced in the Itô sense by multiplicative noise. To get a new trigonometric and hyperbolic stochastic solutions, we use the tanh–coth method and sine–cosine method. In addition, we demonstrate the effect of multiplicative noise on the exact solutions of the stochastic Nizhnik–Novikov–Veselov system

On determining some exact wave solutions to the Nizhnik–Novikov–Veselov system via a rebuts technique

In this study, a significant model of mathematical physics used to describe real-world problems, namely the Nizhnik–Novikov–Veselov (NNV) system is studied and various versions of wave solutions are simultaneously explored. The NNV system is considered an essential model to the KdV hierarchy and is widely used to study the scalar nucleons associated with neutral scalar masons. Thus, the exact wave solutions are important to the associated researchers. To this aim, a robust approach named the generalized exponential rational function method (GERFM) is used to achieve this goal. With the aid of GERFM abundant new wave solutions are formally constructed. All generated solutions are constructed with distinct forms that are dependent on free parameters and determined by the self-form of the NNV system. Moreover, the free parameters play the main roles in the dispersive terms and the values of free parameters influence the physical properties of the traveling solitary and periodic waves to the NNV system directly. Finally, some 3D graphs of the generated solutions are provided to show the propagations of traveling waves. The performed results will have great potential applications to optical fibers, magneto-electrodynamics, the collision of heavy ions, and quantum mechanics.

Using the improved $$\exp ( - \Phi (\xi ))$$ expansion method to find the soliton solutions of the nonlinear evolution equation

With the help of the symbolic calculation software Mathematica, the nonlinear coupled Schrodinger system, (2 + 1)-dimensional dispersive long-wave system and (2 + 1)-dimensional Nizhnik–Novikov–Veselov system are solved by using the improved $$\exp ( - \Phi (\xi ))$$ expansion method. Diverse new soliton solutions are obtained, and the structure and properties of soliton solutions are studied. We derive seven sets of exact solutions under different conditions for the nonlinear coupled Schrodinger system and the (2 + 1)-dimensional dispersive long wave equation, respectively, and for the (2 + 1)-dimensional Nizhnik–Novikov–Veselov system, we find the inelastic interaction solitary waves and fussion soliton.

Rational and semi-rational solutions to the asymmetric Nizhnik–Novikov–Veselov system

In this paper, we use the binary Darboux transformation technique to derive an uniform mathematical expression of all kinds of solutions to the (2 + 1)-dimensional asymmetric Nizhnik–Novikov–Veselov system. For the same seeding solution, a family of eigenfunctions associated with the same eigenvalue is obtained, which is used to construct rational and semi-rational solutions. Interestingly, there exists a category of localized rational solutions that show nontrivial interaction scenarios, namely the pulses undergo a scattering process after the head-on collision. The semi-rational solutions are characterized by two generic evolution scenarios: fission and fusion processes. We also find a subclass of dark rogue waves, namely trains of line solitons that evolve to significant strongly localized transient waves.

Evolution and emergence of new lump and interaction solutions to the (2+1)-dimensional Nizhnik–Novikov–Veselov system

Exploiting Hirota’s bilinear method, we investigate N-soliton solutions, N-order rational solutions, and M-order lump solutions of the (2+1)-dimensional Nizhnik–Novikov–Veselov system. Based on this foundation, different forms of breather wave solutions and lump solutions are obtained by using the parameter limit method. Besides, by constructing a new test function, we study the interaction between lump solutions and soliton solutions of different types, such as the rational-cosh type, rational-cosh-cos type, and rational-cos type. Meanwhile, we also provide a large number of images of the evolution of the spatial structure by selecting different parameter values in order to better show the asymptotic behavior of the exact solution obtained in this paper.

Conservation laws, solitons, breather and rogue waves for the (2+1)-dimensional variable-coefficient Nizhnik–Novikov–Veselov system in an inhomogeneous medium

In this paper, we consider the (2+1)-dimensional variable-coefficient Nizhnik–Novikov–Veselov system in an inhomogeneous medium, which is the isotropic Lax integrable extension of the Korteweg-de Vries equation. Infinitely-many conservation laws are constructed. N-soliton solutions in terms of the Wronskian are obtained via the Hirota method. Velocity and shape of the soliton can be influenced by those variable coefficients, while the amplitude of the soliton can not be affected. Collision between the two solitons is shown to be elastic. Breather wave solutions are constructed via the trilinear method. Such phenomena as the deceleration and compression of the breather waves are studied graphically. Rogue wave solutions are derived when the periods of the breather wave solutions go to the infinity. Separated and united composite rogue waves are discussed graphically.

New solutions for conformable fractional Nizhnik–Novikov–Veselov system via $$G'/G$$ G ′ / G expansion method and homotopy analysis methods

The main purpose of this paper is to find the exact and approximate analytical solution of Nizhnik–Novikov–Veselov system which may be considered as a model for an incompressible fluid with newly defined conformable derivative by using \(G'/G\) expansion method and homotopy analysis method (HAM) respectively. Authors used conformable derivative because of its applicability and lucidity. It is known that, the NNV system of equations is an isotropic Lax integrable extension of the well-known KdV equation and has physical significance. Also, NNV system of equations can be derived from the inner parameter-dependent symmetry constraint of the KP equation. Then the exact solutions obtained by using \(G'/G\) expansion method are compared with the approximate analytical solutions attained by employing HAM.

A new analytical method for seeking traveling wave solutions of space–time fractional partial differential equations arising in mathematical physics

In this paper, based a new fractional sub-equation and the properties of the modified Riemann–Liouville fractional derivative, we propose a new analytical method named improved fractional (DαG/G) method for seeking traveling wave solutions of space–time fractional partial differential equations, which can be seen as the fractional version of the improved (G′/G) method, and is the improvement of the common fractional (DαG/G) method. Due to given traveling wave transformation, one certain space–time fractional partial differential equation can be converted into another fractional ordinary differential equation with respect to one new variable. For demonstrating the validity of this method, we apply it to seek exact traveling wave solutions for the (2+1)-dimensional space–time fractional Nizhnik–Novikov–Veselov System and the space–time fractional KP-BBM equation. With the aid of the mathematical software, some new exact traveling wave solutions including solitary wave solutions, periodic wave solutions and solutions with other forms for them are successfully found.

Nonlocal Symmetry, CRE Solvability and Exact Interaction Solutions of the Asymmetric Nizhnik–Novikov–Veselov System

Abstract Applying the truncated Painlevé expansion to the (2+1)-dimensional asymmetric Nizhnik–Novikov–Veselov (ANNV) system, some Bäcklund transformations (BTs) including auto BT and non-auto BT are obtained. The auto BT leads to a nonlocal symmetry which corresponds to the residual of the truncated Painlevé expansion and the related nonlocal symmetry group is presented with the help of the localization procedure. Further, it is shown that the ANNV system has a consistent Riccati expansion (CRE). Stemming from the consistent tan-function expansion (CTE), which is a special form of CRE, some complex interaction solutions between soliton and arbitrary other seed waves of the ANNV system are readily constructed, such as bight-dark soliton solution, dark-dark soliton solution, soliton-cnoidal interaction solutions, solitoff solutions and so on.

Note on the Equivalence of Variable Separation Solutions Based On the Improved tanh-Function Method

Abstract The equivalence of variable separation solutions based on the improved tanh-function method (ITM) for nonlinear models is illustrated. As an example, we restudy the (2+1)-dimensional generalised Nizhnik–Novikov–Veselov system via the ITM. Based on the radical sign-combined ansatz, five types of variable separation solutions are obtained. By careful analysis, we prove that these seemingly independent variable separation solutions actually depend on each other.

New abundant exact solutions for the (2 + 1)-dimensional generalized Nizhnik–Novikov–Veselov system

In this paper, the extended hyperbolic function method is used for analytic treatment of the (2 + 1)-dimensional generalized Nizhnik–Novikov–Veselov (GNNV) system. We can obtained some new explicit exact solitary wave solutions, the multiple nontrivial exact periodic travelling wave solutions, the soliton solutions and complex solutions. Some known results in the literatures can be regarded as special cases. The methods employed here can also be used to solve a large class of nonlinear evolution equations.

The solution of a coupled system of nonlinear physical problems using the homotopy analysis method

In this article, the homotopy analysis method (HAM) has been applied to solve coupled nonlinear evolution equations in physics. The validity of this method has been successfully demonstrated by applying it to two nonlinear evolution equations, namely coupled nonlinear diffusion reaction equations and the (2+1)-dimensional Nizhnik–Novikov Veselov system. The results obtained by this method show good agreement with the ones obtained by other methods.The proposed method is a powerful and easy to use analytic tool for nonlinear problems and does not need small parameters in the equations. The HAM solutions contain an auxiliary parameter that provides a convenient way of controlling the convergence region of series solutions. The results obtained here reveal that the proposed method is very effective and simple for solving nonlinear evolution equations. The basic ideas of this approach can be widely employed to solve other strongly nonlinear problems.