Maccari Equation Research Articles

General high-order solitons and breathers with a periodic wave background in the nonlocal Hirota–Maccari equation

General high-order solitons and breathers with a periodic wave background in the nonlocal Hirota–Maccari equation

Rogue Waves Generator and Chaotic and Fractal Behavior of the Maccari System with a Resonant Parametric Forcing

Using the Asymptotic Perturbation (AP) method we can find approximate solutions for the Maccari equation with a parametric resonant forcing acting over the frequency of a generic mode. Taking into account its nonlocal behavior and applying symmetry considerations, a system with two coupled equations for the phase and amplitude modulation can be obtained. The system can be solved, and we demonstrate the existence of a big modulation in the wave amplitude, producing a rogue waves train and, in this case, these waves are not isolated. We then obtain a rogue waves generator, being able of producing and controlling the rogue waves’ amplitude. Another important finding is the existence of chaotic or fractal solutions, because of the presence of an arbitrary function in the solution.

A Parametric Resonance for the Nonlocal Hirota–Maccari Equation

The nonlocal Hirota–Maccari equation is considered when a parametric excitation is acting over the frequency of a generic mode. Using the well-known asymptotic perturbation (AP) method, two coupled equations for the amplitude and phase can be obtained. We discovered the existence of an infinite-period bifurcation when the parametric force increases its value. Moreover, symmetry considerations suggest performing a global analysis of the two couples, in such a way that we find an energy-like function and corroborate and verify the existence of this infinite period bifurcation.

The soliton solutions of (2 1)-dimensional nonlinear two-coupled Maccari equation with complex structure via new Kudryashov scheme

The soliton solutions of (2 1)-dimensional nonlinear two-coupled Maccari equation with complex structure via new Kudryashov scheme

Traveling Wave Solutions to the Nonlinear Evolution Equation Using Expansion Method and Addendum to Kudryashov’s Method

The inspection of wave motion and propagation of diffusion, convection, dispersion, and dissipation is a key research area in mathematics, physics, engineering, and real-time application fields. This article addresses the generalized dimensional Hirota–Maccari equation by using two different methods: the exp(−φ(ζ)) expansion method and Addendum to Kudryashov’s method to obtain the optical traveling wave solutions. By utilizing suitable transformations, the nonlinear pdes are transformed into odes. The traveling wave solutions are expressed in terms of rational functions. For certain parameter values, the obtained optical solutions are described graphically with the aid of Maple 15 software.

On the exact solutions to some system of complex nonlinear models

Abstract In this manuscript, the application of the extended sinh-Gordon equation expansion method to the Davey-Stewartson equation and the (2+1)-dimensional nonlinear complex coupled Maccari system is presented. The Davey-Stewartson equation arises as a result of multiple-scale analysis of modulated nonlinear surface gravity waves propagating over a horizontal seabed and the (2+1)-dimensional nonlinear complex coupled Maccari equation describes the motion of the isolated waves, localized in a small part of space, in many fields such as hydrodynamic, plasma physics, nonlinear optics. We successfully construct some soliton, singular soliton and singular periodic wave solutions to these two nonlinear complex models. The 2D, 3D and contour graphs to some of the obtained solutions are presented.

Abundant soliton solutions for the Hirota–Maccari equation via the generalized exponential rational function method

In this paper, some new traveling wave solutions to the Hirota–Maccari equation are constructed with the help of the newly introduced method called generalized exponential rational function method. Several families of exact solutions are found corresponding to the equation. To the best of our knowledge, these solutions are new, and have never been addressed in the literature. The graphical interpretation of the solutions is also depicted. Moreover, it is contemplated that the proposed technique can also be employed to another sort of complex models.

M-fractional solitons and periodic wave solutions to the Hirota–Maccari system

In this study, we construct several wave solutions to the nonlinear fractional Hirota–Maccari equation with a truncated M-fractional derivative via the extended sinh-Gordon equation expansion method. The constraint conditions that guarantee the existence of valid solutions are stated. We use suitable values of parameters in plotting the 2- and 3-dimensional graphs of the reported solutions.

Homoclinic breather and rogue wave solutions to Maccari equation

In this work, a new extended homoclinic test combining with breather period limit method (BPLM), for seeking homoclinic breather and rogue wave solution of nonlinear evolution (2+1)D Maccari equation, is proposed. Two new families of homoclinic breather wave solution and rational homoclinic solution (homoclinic rogue wave) for these equations are obtained, respectively. Moreover, dynamical features of homoclinic breather and rogue wave solution are exhibited. This result shows that both homoclinic breather and rogue wave can occur in the same nonlinear system.

Dispersive solitary wave solutions of new coupled Konno-Oono, Higgs field and Maccari equations and their applications

In this research we apply generalized Exp-Function method to obtain exact, solitary and new soliton wave solutions of new coupled Konno-Oono equation, Higgs field equation and Maccari equation via generalized Exp-Function method which are very substantial models in define a current-fed string interacting with an external magnetic field in three-dimensional Euclidean space, introduces quantum field (or the Higgs field) to illustrate the generation mechanism of mass for gauge bosons and described the motion of the isolated waves, localized in a small part of space, in many fields such as hydrodynamic, plasma physics, nonlinear optics and others. generalized Exp-Function method is very sturdy, fabulous, felicitous and effective method to get exact, solitary and new soliton wave solution of nonlinear partial differential equations (PDEs.). We present a contrasting between the results of this modern method and another method and show that how the results that obtained by this method is much closed to cover many different methods in this field and not just that but also get a new solitary and soliton wave solutions which give a wide range of solutions that help all researchers who apply these models in our life.

Exact traveling wave solutions of some nonlinear evolution equations

Using a traveling wave reduction technique, we have shown that Maccari equation, (2?1)-dimensional nonlinear Schrodinger equation, medium equal width equation, (3?1)-dimensional modified KdV-Zakharov- Kuznetsev equation, (2?1)-dimensional long wave-short wave resonance interaction equation, perturbed nonlinear Schrodinger equation can be reduced to the same family of auxiliary elliptic-like equations. Then using extended F-expansion and projective Riccati equation methods, many types of exact traveling wave solutions are obtained. With the aid of solutions of the elliptic-like equation, more explicit traveling wave solutions expressed by the hyper- bolic functions, trigonometric functions and rational func- tions are found out. It is shown that these methods provide a powerful mathematical tool for solving nonlinear evolu- tion equations in mathematical physics. A variety of structures of the exact solutions of the elliptic-like equation are illustrated.

Bifurcations of Exact Traveling Wave Solutions of Maccari's Equations

For the Maccari's equations, the objective of this paper is to investigate the dynamical behavior of its traveling wave solutions by using the bifurcation method and qualitative theory of dynamical systems. All exact explicit parametric respresentations of solitary wave, kink (anti-kink) wave and periodic wave solutions are obtained under given parameter conditions, and the dynamics characters of these solutions are also analyzed.

Exact travelling wave solutions for some important nonlinear physical models

The two-dimensional nonlinear physical models and coupled nonlinear systems such as Maccari equations, Higgs equations and Schrodinger–KdV equations have been widely applied in many branches of physics. So, finding exact travelling wave solutions of such equations are very helpful in the theories and numerical studies. In this paper, the Kudryashov method is used to seek exact travelling wave solutions of such physical models. Further, three-dimensional plots of some of the solutions are also given to visualize the dynamics of the equations. The results reveal that the method is a very effective and powerful tool for solving nonlinear partial differential equations arising in mathematical physics.

N-soliton solutions for the [formula omitted]-dimensional Hirota–Maccari equation in fluids, plasmas and optical fibers

Under investigation in this paper is the Hirota–Maccari equation, which is a generalized ( 2 + 1 ) -dimensional model in fluid dynamics, plasma physics and optical fiber communication. With the aid of the Hirota bilinear method and symbolic computation, the corresponding N-soliton solutions are given and illustrated. The characteristic face method and asymptotic analysis are applied to discuss the solitonic propagation and collision, including the bidirectional solitons, elastic interactions and inelastic interactions. Finally, a kind of special phenomenon with the parameters varying is investigated, which might provide people with useful information on the dynamics of the relevant fields.

New exact travelling wave solutions of some complex nonlinear equations

In this paper, we establish exact solutions for complex nonlinear equations. The tanh–coth and the sine–cosine methods are used to construct exact periodic and soliton solutions of these equations. Many new families of exact travelling wave solutions of the coupled Higgs and Maccari equations are successfully obtained. These solutions may be important of significance for the explanation of some practical physical problems.

A direct algebraic method applied to obtain complex solutions of some nonlinear partial differential equations

By using some exact solutions of an auxiliary ordinary differential equation, a direct algebraic method is described to construct the exact complex solutions for nonlinear partial differential equations. The method is implemented for the NLS equation, a new Hamiltonian amplitude equation, the coupled Schrodinger–KdV equations and the Hirota–Maccari equations. New exact complex solutions are obtained.

NEW TYPES OF INTERACTIONS BASED ON VARIABLE SEPARATION SOLUTIONS VIA THE GENERAL PROJECTIVE RICCATI EQUATION METHOD

In this paper, first, the general projective Riccati equation method (PREM) is applied to derive variable separation solutions of (2 + 1)-dimensional systems. By further studying, we find that these variable separation solutions obtained by PREM, which seem independent, actually depend on each other. A common formula with some arbitrary functions is obtained to describe suitable physical quantities for some (2 + 1)-dimensional models such as the generalized Nizhnik–Novikov–Veselov system, Broer–Kaup–Kupershmidt equation, dispersive long wave system, Boiti–Leon–Pempinelli model, generalized Burgers model, generalized Ablowitz–Kaup–Newell–Segur system and Maccari equation. The universal formula in Tang, Lou, and Zhang [2] can be simplified to the common formula in the present paper. Second, this method is successfully generalized to (1 + 1)-dimensional systems, such as coupled integrable dispersionless equations, shallow water wave equation, Boiti system and negative KdV model, and is able to obtain another common formula to describe suitable physical fields or potentials of these (1 + 1)-dimensional models, which is similar to the one in (2 + 1)-dimensional systems. Finally, based on the common formula for (2 + 1)-dimensional systems and by selecting appropriate multivalued functions, elastic and inelastic interactions among special dromion, special peakon, foldon and semi-foldon are investigated. Furthermore, the explicit phase shifts for all the local excitations offered by the common formula have been given, and are applied to these novel interactions in detail.

Complex hyperbolic-function method and its applications to nonlinear equations

Based on computerized symbolic computation, a complex hyperbolic-function method is proposed for the general nonlinear equations of mathematical physics in a unified way. In this method, we assume that exact solutions for a given general nonlinear equations be the superposition of different powers of the sech-function, tanh-function and/or their combinations. After finishing some direct calculations, we can finally obtain the exact solutions expressed by the complex hyperbolic function. The characteristic feature of this method is that we can derive exact solutions to the general nonlinear equations directly without transformation. Some illustrative equations, such as the ( 1 + 1 )-dimensional coupled Schrödinger–KdV equation, ( 2 + 1 )-dimensional Davey–Stewartson equation and Hirota–Maccari equation, are investigated by this means and new exact solutions are found.