Dimensional Nonlinear Equation Research Articles

Soliton solutions for a (3 + 1)-dimensional nonlinear integrable equation

Soliton solutions for a (3 + 1)-dimensional nonlinear integrable equation

Diverse optical soliton solutions of two space-time fractional nonlinear evolution equations by the extended kudryashov method

This study investigates the inclusive optical soliton solutions to the (2+1)-dimensional nonlinear time-fractional Zoomeron equation and the space-time fractional nonlinear Chen-Lee-Liu equation using the extended Kudryashov technique. The beta derivative is used to conduct the fractional terms and investigate wide-spectral soliton solutions to the considered models. The obtained solutions yield a variety of typical soliton shapes, including ant-peakon soliton, V-shaped soliton, anti-bell-shaped soliton, kink soliton, periodic soliton, singular periodic soliton for the specific value of the parameters. The three-dimensional, contour, and two-dimensional graphs of the derived solitons have been plotted to illustrate the structure, propagation, and influence of the fractional parameter. It is observed that the fractional parameter affects the amplitudes and periods of certain solitons. The precision of the acquired solutions is confirmed by reintroducing them into the original equation using Mathematica. The findings of this study indicate that the employed method has the capability of yielding compatible, creative, and useful solutions for diverse nonlinear evolution equations with fractional derivatives. This approach could introduce novel ways for unraveling other nonlinear equations and have implications in diverse sectors of nonlinear science and engineering.

Exploring lump soliton solutions and wave interactions using new Inverse $$(G'/G)$$-expansion approach: applications to the (2+1)-dimensional nonlinear Heisenberg ferromagnetic spin chain equation

Exploring lump soliton solutions and wave interactions using new Inverse $$(G'/G)$$-expansion approach: applications to the (2+1)-dimensional nonlinear Heisenberg ferromagnetic spin chain equation

Propagation of electrostatic potential with dynamical behaviors and conservation laws of the (3+1)-dimensional nonlinear extended quantum Zakharov–Kuznetsov equation

The [Formula: see text]-expansion scheme is used to execute many wave results for the partial differential equation, namely, the nonlinear extended quantum Zakharov–Kuznetsov (NLEQZK) equation which plays a significant role in mathematical physics, and this equation is accomplished in quantum electron–positron–ion magneto plasmas. The Lie approach is used to find the infinitesimal generators, and group invariant solutions, and help us to reduce the considered PDEs into ODEs. Then a planer dynamical system approach is used to see the existence of closed-form solutions. All possible phase portraits are obtained and the existence of electrostatic wave potentials is reported. Meantime, periodic and super-nonlinear electrostatic wave potentials are found for different initial value conditions with the help of the Runge–Kutta method. Moreover, new traveling wave patterns of the considered models are constructed and presented graphically. Then, the conserved vectors of the given physical model with the use of the multiplier scheme are presented.

Global existence and spatial analyticity for a nonlocal flux with fractional diffusion

In this paper, we study a one dimensional nonlinear equation with diffusion −ν(−∂xx)α2 for 0 ≤ α ≤ 2 and ν > 0. We use a viscous-splitting algorithm to obtain global nonnegative weak solutions in space L1(R)∩H1/2(R) when 0 ≤ α ≤ 2. For the subcritical case 1 < α ≤ 2 and critical case α = 1, we obtain the global existence and uniqueness of nonnegative spatial analytic solutions. We use a fractional bootstrapping method to improve the regularity of mild solutions in the Bessel potential spaces for the subcritical case 1 < α ≤ 2. Then, we show that the solutions are spatial analytic and can be extended globally. For the critical case α = 1, if the initial data ρ0 satisfies −ν < inf ρ0 < 0, we use the method of characteristics for complex Burgers equation to obtain a unique spatial analytic solution to our target equation in some bounded time interval. If ρ0 ≥ 0, the solution exists globally and converges to steady state.

Micro-polar liquid flows with magnetohydrodynamic (MHD) mixed convection due to a nonlinear stretched sheet in a convective state

The topic of this study is the flow of a micro-polar liquid toward a nonlinear stretched surface using magnetohydrodynamics (MHDs). The analysis presents the effects of some important parameters on fluid flow. The mixed convection phenomena are used to examine the characteristics of heat transmission. By using the appropriate transformations, dimensional nonlinear equations can be transformed into dimensionless expressions. The governing dimensionless problems are solved using the modified Laplace decomposition approach by examining the effects of various boundary factors on flow and heat transfer.

Dynamics of breather, multi-wave, interaction and other wave solutions to the new (3+1)-dimensional integrable fourth-order equation for shallow water waves

PurposeThe purpose of this paper is to study the new (3 + 1)-dimensional integrable fourth-order nonlinear equation which is used to model the shallow water waves.Design/methodology/approachBy means of the Cole–Hopf transform, the bilinear form of the studied equation is extracted. Then the ansatz function method combined with the symbolic computation is implemented to construct the breather, multiwave and the interaction wave solutions. In addition, the subequation method tis also used to search for the diverse travelling wave solutions.FindingsThe breather, multiwave and the interaction wave solutions and other wave solutions like the singular periodic wave structure and dark wave structure are obtained. To the author’s knowledge, the solutions obtained are all new and have never been reported before.Originality/valueThe solutions obtained in this work have never appeared in other literature and can be regarded as an extension of the solutions for the new (3 + 1)-dimensional integrable fourth-order nonlinear equation.

Bilinear auto-Bäcklund transformations, breather and mixed lump–kink solutions for a (3+1)-dimensional integrable fourth-order nonlinear equation in a fluid

In this paper, we investigate a (3+1)-dimensional integrable fourth-order nonlinear equation that can model both the right- and left-going waves in a fluid. Based on the Hirota method, we derive three sets of the bilinear auto-Bäcklund transformations along with some analytic solutions. Through the extended homoclinic test approach, we obtain some breather solutions. We find that the breather propagates steadily along a straight line, with one hole and one peak in each period. Graphical investigation indicates that the coefficients in that equation affect the location and shape of the breather. Moreover, we construct some mixed lump–kink solutions. Fusion and fission between a lump wave and a kink soliton are analyzed graphically. The solutions addressed in this paper may be applied to mimic some complex waves in fluids.

Searching For (2+1)-dimensional nonlinear Boussinesq equation from (1+1)-dimensional nonlinear Boussinesq equation

A novel (2+1)-dimensional nonlinear Boussinesq equation is derived from a (1+1)-dimensional Boussinesq equation in nonlinear Schrödinger type based on a deformation algorithm. The integrability of the obtained (2+1)-dimensional Boussinesq equation is guaranteed by its Lax pair obtained directly from the Lax pair of the (1+1)-dimensional Boussinesq equation. Because of the effects of the deformation, the (2+1)-dimensional Boussinesq equation admits a special travelling wave solution with a shape that can be deformed to be asymmetric and/or multi-valued.

Solution Structures of an Electrical Transmission Line Model with Bifurcation and Chaos in Hamiltonian Dynamics

Employing the Riccati–Bernoulli sub-ODE method (RBSM) and improved Weierstrass elliptic function method, we handle the proposed [Formula: see text]-dimensional nonlinear fractional electrical transmission line equation (NFETLE) system in this paper. An infinite sequence of solutions and Weierstrass elliptic function solutions may be obtained through solving the NFETLE. These new exact and solitary wave solutions are derived in the forms of trigonometric function, Weierstrass elliptic function and hyperbolic function. The graphs of soliton solutions of the NFETLE describe the dynamics of the solutions in the figures. We also discuss elaborately the effects of fraction and arbitrary parameters on a part of obtained soliton solutions which are presented graphically. Moreover, we also draw meaningful conclusions via a comparison of our partially explored areas with other different fractional derivatives. From our perspectives, by rewriting the equation as Hamiltonian system, we study the phase portrait and bifurcation of the system about NFETLE and we also for the first time discuss sensitivity of the system and chaotic behaviors. To our best knowledge, we discover a variety of new solutions that have not been reported in existing articles [Formula: see text], [Formula: see text]. The most important thing is that there are iterative ideas in finding the solution process, which have not been seen before from relevant articles such as [Tala-Tebue et al., 2014; Fendzi-Donfack et al., 2018; Ashraf et al., 2022; Ndzana et al., 2022; Halidou et al., 2022] in seeking for exact solutions about NFETLE.

Soliton solutions to generalized (2+1)-dimensional Hietarinta-type equation and resonant NLSE along with stability analysis

This paper discusses the generalized [Formula: see text]-dimensional Hietarinta-type fourth-order nonlinear equation (GHTE) and Resonant nonlinear Schrödinger equation (RNLSE). We will study the soliton solutions of the GHTE as well as RNLSE by the renowned, efficient and successful method called sub-ODE technique. The solutions are categorized as dark soliton, periodic, rational, positive, hyperbolic and Weierstrass elliptic solutions (WES). Stability analysis of the solutions is performed for both the models. Later on, these results will be displayed graphically by 3D, 2D, contour, density and streamline plots which describe the behavior of the waves.

Time-Dependent Magnetohydrodynamic (MHD) Flow of an Exothermic Arrhenius Fluid in a Vertical Channel with Convective Boundary Condition

The current study examined the effects of magnetohydrodynamics (MHD) on time-dependent mixed convection flow of an exothermic fluid in a vertical channel. Convective heating and Navier’s slip conditions are considered. The dimensional nonlinear flow equations are transformed into dimensionless form with suitable transformation. For steady-state flow formations, we apply homotopy perturbation approach. However, for the unsteady-state governing equation, we use numerical technique known as the implicit finite difference approach. Flow is influenced by several factors, including the Hartmann number, Newtonian heating, Navier slip parameter, Frank-Kamenetskii parameter, and mixed convection parameter. Shear stress and heat transfer rates were also investigated and reported. The steady-state and unsteady-state solutions are visually expressed in terms of velocity and temperature profiles. Due to the presence of opposing force factors such as the Lorentz force, the research found that the Hartmann number reduces the momentum profile. Fluid temperature and velocity increase as the thermal Biot number and Frank-Kamenetskii parameter increase. There are several scientific and infrastructure capabilities that use this type of flow, such flow including solar communication systems exposed to airflow, electronic devices cooled at room temperature by airflow, nuclear units maintained during unscheduled shutoffs, and cooling systems occurring in low circumstances. The current findings and the literature are very consistent, which recommend the application of the current study.

Controllable rogue waves in a compressible hyperelastic plate

In this paper, we investigate various rational solutions of a (2+1)-dimensional nonlinear equation with variable-coefficients which can describe the propagation of nonlinear waves in a compressible hyperelastic plate, and interestingly we find a new phenomenon in nonlinear hyperelastic mechanics—rogue waves. Based on the symmetry transformation and Hirota bilinear form, we obtain breather waves and rogue waves. And when we choose appropriate material parameters and transformation functions, we can control their directions of propagation. In addition, we present three new interaction solutions, i.e. the rogue waves keep moving from side to side between a pair of resonance stripe solitary waves. Finally, we analysis the dynamics characteristics and evolution behaviors of the obtained solutions by three-dimensional dynamic images. This paper exhibits the existence of rogue waves in nonlinear hyperelastic mechanics, and presents potential application value for control of rogue waves in elastic mechanics, fluid dynamics and other fields of nonlinear science.

Efficient SAV–Hermite methods for the nonlinear Dirac equation

We construct two SAV/CN–Hermite schemes for one and two dimensional nonlinear Dirac equation in this paper. Both SAV/CN methods preserve the energy and mass and an efficient and accurate scheme is proposed by Hermite approximation. The numerical analysis of error estimates is established on SAV/CN scheme. Numerical experiments confirm the expected convergence order. In addition, they are given to show the simulations of binary and ternary collisions.

Lie symmetry analysis and exact solution of $$(2+1)$$-dimensional nonlinear time-fractional differential-difference equations

Lie symmetry analysis and exact solution of $$(2+1)$$-dimensional nonlinear time-fractional differential-difference equations

Abundant invariant and classical solutions with the conservation laws of a new (3+1)-dimensional fifth-order nonlinear Wazwaz equation with the third-order dispersion terms in ocean physics

Ocean plays key roles in the earth’s system in shaping the planet’s climate as well as weather by absorbing, storing, and also transporting huge quantities of carbon-dioxide, water, heat and moisture. This paper presents the analytical examination of a new (3+1)-dimensional fifth-order nonlinear Wazwaz equation with third-order dispersion terms which is applicable in ocean physics and other nonlinear sciences. Ocean physics is the study of various conditions as well as processes that are physical within the ocean, especially the physical features and motions of ocean waters. Thus, the diverse outcomes of the investigations carried out on the model under study possesses significant applications in physical oceanography (Adeyemo, 2022). We first apply Lie group analysis to obtain various infinitesimal generators admitted by the equation. This is followed by invoking the generators to reduce the understudy equation for the purpose of achieving copious group-invariant solutions. We solve the resultant ordinary differential equations in order to obtain diverse classical solutions of the underlying equation. The outcome yields several solutions with arbitrary functions, quadratures, rational and hyperbolic functions alongside various topological kink solitons. Moreover, numerical simulations of the obtained results are performed so as to give some physical interpretations to the solutions and these yield various solitary waves as well as interactions between soliton waves of interest. Furthermore, we outline the applications of our results in ocean physics. Conclusively, conservation laws are constructed for the equation understudy with the use of Ibragimov’s theorem.

Solutions of Three Dimensional Nonlinear Klein-Gordon Equations by Using Quadruple Laplace Transform

This study focuses on solving three-dimensional non-linear Klein-Gordon equations of four variables by using the quadruple Laplace transform method coupled with the iterative method. This study was designed in order to show the quadruple Laplace transform with an iterative method for solving three-dimensional nonlinear Klein-Gordon equations. The quadruple Laplace transform with the iterative method was aimed at getting analytical solutions of three-dimensional nonlinear Klein-Gordon equations. Exact solutions obtained through the iterative method have been analytically evaluated and presented in the form of a table and graph. The analytical solutions of these equations have been given in terms of convergent series with a simply calculable system, and the nonlinear terms in equations can easily be solved by the iterative method. Illustrative examples are also provided to demonstrate the applicability and efficiency of the method. The result renders the applicability and efficiency of the applied method. Finally, the quadruple Laplace transform and iterative method is an excellent method for the solution of nonlinear Klein-Gordon equations.

Efficient discrete singular convolution differential quadrature algorithm for solitary wave solutions for higher dimensions in shallow water waves

In this research, discrete singular convolution, that depends on Regularized Shannon kernel, is used to look for the efficient solution of (4 + 1) dimensional nonlinear Fokas equation. The governing system of nonlinear five-dimensional Fokas equation is transformed into a system of nonlinear ordinary differential equations via discrete singular convolution quadrature technique. Consequently, the 4th order Runge–Kutta technique is employed to solve this system. The validity of this approach is achieved by comparing the obtained numerical solution with the exact solution, where the error is ≤ 1 × 10−5. Moreover, a parametric analysis is presented to examine the influence of the used approach on solitary wave solution. The computed solutions display strength, reliability and efficiency of the implemented techniques that can be effectively work for nonlinear equations in a like manner to linear models. Our treatment in this work works to solve higher dimensional nonlinear problems of physical and numerical features.

Soliton solutions to the nonlinear higher dimensional Kadomtsev-Petviashvili equation through the new Kudryashov’s technique

In this paper, we studied the (3 + 1)-dimensional nonlinear Kadomtsev-Petviasvili equation (3D-KPE) that is utilized in order to describe 3D solitons in weakly dispersive media, long wavelength water waves with weak nonlinear restoring forces, waves in ferromagnetic media, nonlinear wave propagation in supefluids, plasma physics and fluid dynamics by using the recently presented the new Kudryashov’s method. We successfully applied the new Kudryashov’s scheme to the investigated problem for the first time to achieve bright and singular soliton; besides, we showed that the technique is effective, easily applicable, and reliable in solving such nonlinear problems. Moreover, the necessary comments were given by obtaining appropriate soliton solutions and presented 3D and 2D graphics.

Diverse Soliton Structures of the ( 2 + 1 )-Dimensional Nonlinear Electrical Transmission Line Equation

In this work, the ( 2 + 1 )-dimensional nonlinear electrical transmission line equation (NETLE) is investigated by applying three recent technologies, namely, the variational approach, Hamiltonian approach, and energy balance approach. Diverse exact soliton solutions such as the bright, bright-like, kinky bright, bright-dark soliton, and periodic soliton solutions are successfully constructed. The outlines of the different solutions are shown in the form of the 3-D plot with the help of the Wolfram Mathematica. It reveals that the used methods are concise and effective and are expected to provide some inspiration for the study of travelling wave solutions of the PDEs in physics.