Asymmetric Nizhnik Novikov Veselov 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.

Exploring plasma phenomena with the Nizhnik–Novikov–Veselov formula: Analyzing ion-acoustic waves, solitons, and shocks

This paper delves into the intricacies of the ([Formula: see text])-dimensional asymmetric Nizhnik–Novikov–Veselov ([Formula: see text]) model, a nonlinear partial differential equation governing ion-acoustic wave propagation in plasma. By employing advanced analytical and numerical approaches, the study explores innovative solitary wave solutions, particularly focusing on the dynamics of isochoric flow. Isochoric flow analysis is crucial for unraveling the complex behaviors exhibited by incompressible fluids like elastomers and bio-elastomers, which maintain a constant density. The derivation of the ([Formula: see text])-dimensional [Formula: see text] equation stems from fluid equations governing plasma dynamics. This model serves as a valuable tool for simulating experimental observations of plasma waves. The computational methodology applied in this research demonstrates a commendable level of precision and consistency, yielding novel solitary wave solutions previously unreported in the [Formula: see text] model. These results underscore the study’s importance and novelty. The outcomes not only contribute to our understanding of incompressible fluid dynamics, but also lay the groundwork for future investigations in this domain. The revealed solitary wave solutions have the potential to inform the development of more accurate models for predicting fluid dynamics, thereby advancing the field.

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.

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.

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.

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.

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.

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.

Dynamical interactions between higher-order rogue waves and various forms of n-soliton solutions (n → ∞) of the (2+1)-dimensional ANNV equation

We present new lemmas, theorem and corollaries to construct interactions among higher-order rogue waves, n-periodic waves and n-solitons solutions (n → ∞) to the (2+1)-dimensional asymmetric Nizhnik–Novikov–Veselov (ANNV) equation. Several examples for theories are given by choosing definite interactions of the wave solutions for the model. In particular, we exhibit dynamical interactions between a rogue and a cross bright-dark bell wave, a rogue and a cross-bright bell wave, a rogue and a one-, two-, three-, four-periodic wave. In addition, we also present multi-types interactions between a rogue and a periodic cross-bright bell wave, a rogue and a periodic cross-bright-bark bell wave. Finally, we physically explain such interaction solutions of the model in the 3D and density plots.

Complexiton solutions to the asymmetric Nizhnik–Novikov–Veselov equation

We present new complexiton solutions to the (2 + 1)-dimensional asymmetric Nizhnik–Novikov–Veselov (aNNV) equation by application of the Hirota direct method and the linear superposition principle. We first find hyperbolic function solutions to the corresponding bilinear equation and consequently derive the so-called complexitons. In particular, we construct nonsingular complexiton solutions from positive complexiton solutions of the bilinear form of the nonlinear equation. Finally, we give some illustrative examples and a few concluding remarks.

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.

Consistent Riccati Expansion for Integrable Systems

A consistent Riccati expansion (CRE) is proposed for solving nonlinear systems with the help of a Riccati equation. A system having a CRE is then defined to be CRE solvable. The CRE solvability is demonstrated quite universal for various integrable systems including the Korteweg–de Vries, Kadomtsev–Petviashvili, Ablowitz–Kaup–Newell–Segur (and then nonlinear Schrödinger), sine‐Gordon, Sawada–Kotera, Kaup–Kupershmidt, modified asymmetric Nizhnik–Novikov–Veselov, Broer–Kaup, dispersive water wave, and Burgers systems. In addition, it is revealed that many CRE solvable systems share a similar determining equation describing the interactions between a soliton and a cnoidal wave. They have a common nonlocal symmetry expression and they possess a formally universal once Bäcklund transformation.

Bifurcations of traveling wave solutions for the (2 + 1)-dimensional generalized asymmetric Nizhnik–Novikov–Veselov equation

By using the bifurcation theory of planar dynamical systems to the (2+1)-dimensional generalized asymmetric Nizhnik–Novikov–Veselov equation, the existence for solitary wave solutions and uncountably infinite many smooth and non-smooth periodic wave solutions is obtained. Under different regions of parametric spaces, various sufficient conditions to guarantee the existence of these solutions mentioned are given. Furthermore, some exact explicit parametric expressions of these bounded traveling waves are obtained.

Special Bi-Solitons for Asymmetric Nizhnik—Novikov—Veselov Equation

Employing a constructive algorithm and the symbolic computation, we obtain a new explicit bi-soliton-like solution of the asymmetric Nizhnik—Novikov—Veselov equation. The solution contains two arbitrary functions which indicates that it can model various bi-soliton-like waves. In particular, specially choosing the arbitrary functions, we find some interesting bi-solitons with special shapes, which possess the traveling property of the traditional bi-solitons. We show the evolution of such bi-solitons by figures.

New application to Riccati equation

To seek new infinite sequence of exact solutions to nonlinear evolution equations, this paper gives the formula of nonlinear superposition of the solutions and Bäcklund transformation of Riccati equation. Based on the tanh-function expansion method and homogenous balance method, new infinite sequence of exact solutions to Zakharov–Kuznetsov equation, Karamoto–Sivashinsky equation and the set of (2+1)-dimensional asymmetric Nizhnik–Novikov–Veselov equations are obtained with the aid of symbolic computation system Mathematica. The method is of significance to construct infinite sequence exact solutions to other nonlinear evolution equations.

A new algebraic procedure to construct exact solutions of nonlinear differential–difference equations

This paper presents a new algebraic procedure to construct exact solutions of selected nonlinear differential–difference equations. The discrete sine-Gordon equation and differential–difference asymmetric Nizhnik–Novikov–Veselov equations are chosen as examples to illustrate the efficiency and effectiveness of the new procedure, where various types of exact travelling wave solutions for these nonlinear differential–difference equations have been constructed. It is anticipated that the new procedure can also be used to produce solutions for other nonlinear differential–difference equations.

Multiple soliton solutions for the ( [formula omitted])-dimensional asymmetric Nizhnik–Novikov–Veselov equation

The asymmetric Nizhnik–Novikov–Veselov equation will be investigated. The Hirota bilinear method will be used to derive N -soliton solutions of this completely integrable equation. Multiple singular soliton solutions will be developed as well. The asymmetric Nizhnik–Novikov–Veselov equation does not show any resonant phenomenon.

New variable separation solutions of the [formula omitted]-dimensional asymmetric Nizhnik–Novikov–Veselov system

By means of the exp-function method, we obtain some new exact solutions of the Riccati equation. Based on the Riccati equation and its new exact solutions, we find new and more general variable separation solutions with two arbitrary functions of the (2+1)-dimensional asymmetric Nizhnik–Novikov–Veselov (ANNV) system. As some special examples, these new variable separation solutions can degenerate into some known solutions reported in open literature.

New Exact Solutions and Complex Wave Excitations for the (2+1)-Dimensional Asymmetric Nizhnik-Novikov-Veselov System

Starting from an improved mapping approach and a linear variable separation approach, new families of variable separated solutions (including solitary wave solutions, periodic wave solutions and rational function solutions) with arbitrary functions for the (2+1)-dimensional asymmetric Nizhnik- Novikov-Veselov (ANNV) system are derived. Based on the derived solutions, we obtain some special complex wave excitations.