Dimensional Equation Research Articles

Nonlocal symmetry and group invariant solutions of dissipative (2+1)-dimensional AKNS equation

In this paper, the nonlocal symmetry of dissipative (2+1)-dimensional AKNS equation is constructed by auxiliary equation method. In order to better study new group invariant solutions of AKNS equation with the help of nonlocal symmetry, we introduce an new auxiliary dependent variable, the (2+1) dimensional AKNS equation is extended to a new closed prolonged system. Therefore, the nonlocal symmetry of the original equation is the Lie point symmetry of the new prolonged system. Then, by solving the initial-value problems, the finite symmetry transformation for the prolonged system is derived. Furthermore, by using symmetry reduction method to the prolonged system, the interaction solution between cnoidal waves and solitary waves is given.

Breather waves, periodic cross-lump waves and complexiton type solutions for the (2 + 1)-dimensional Kadomtsev-Petviashvili equation in dispersive media

The manuscript under consideration explores the (2+1)-dimensional Kadomtsev-Petviashvili (KP) model using its bilinear representation. The focus is on studying the characteristics of periodic cross-lump waves and breather wave solutions. Breathers, which are localized periodic solutions of continuous media equations or discrete lattice equations, are of particular interest. The research also delves into complexiton solutions using the extended transformed rational function approach. The study graphically analyzes the dynamics of these interaction solutions using appropriate parameters. It's truly fascinating to learn about the application of breathers in continuous media equations or discrete lattice equations.

Decomposition and linear superposition of the (2+1)-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada equation

Generally, the principle of linear superposition is not applicable in nonlinear systems. The results of our researches prove that for some certain types of decomposed solutions of (2+1)-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada(CDGKS) equation, the linear superposition principle holds. First, beginning with the potential (2+1)-dimensional CDGKS equation, six sets of decomposition equations are formulated by the formally variable separation approach. On the basis of these decompositions, the linear superposition solutions are then discussed. More specifically, the combination of two periodic wave solutions, soliton and periodic wave interaction solution and the multi-soliton solution are obtained.

Viscous dissipation effect on amplitude and oscillating frequency of heat transfer and electromagnetic waves of magnetic driven fluid flow along the horizontal circular cylinder

The significant importance of present research is to remove the extreme temperature along the magnetic driven horizontal circular cylinder. The induced electromagnetic field is applied around the surface of cylinder. The main novelty of current research is to control thermal and magnetic boundary layer in the presence of viscous dissipation and induced electromagnetic field. The dimensional mathematical form is developed with defined boundary conditions. The dimensional equations are transformed into dimensionless equations to generate physical factors. The primitive form is used to reduce dimensionless equations into convenient form for smooth algorithm. The finite difference method with Gaussian elimination technique is applied for numerical results in FORTRAN language tool. The velocity, temperature and electromagnetic field are sketched graphically with asymptotic sequence. The oscillatory shear stress, oscillating heat rate and periodical current density is plotted graphically and numerically. It is found that fluid velocity improves significantly as buoyancy force increases around each position. It is noticed that the increasing oscillations in heat transfer are sketched for maximum choice of Prandtl number. It is found that the maximum oscillations in current density are obtained for each Eckert parameter. It is noticed that the significant distribution in temperature profile is obtained in the presence of viscous dissipation and magnetic field.

Families of exact solutions of a Generalized (2+1)-dimensional Boussinesq type equation

In this paper, we study a Generalized (2+1)-dimensional Boussinesq-type equation. Using the Hirota bilinear method, we present the N-order bright soliton solutions and dark soliton solutions. For the one-soliton solution, the bright soliton solution and the dark soliton solution share the same limit line but have different extreme values. Building on the soliton solutions, we derive higher-order bright and dark breather solutions as well as mixed solutions. The dynamic behavior is characterized using visual representations. Furthermore, through the long-wave limit method, we obtain the bright and dark lump solutions. Notably, they share the same extreme points but have different extreme values. Additionally, we derive two semi-rational solutions as lump-soliton and lump-breather. It is found that the lump moves along the peak amplitude of the soliton wave.

Dynamics of transformed nonlinear waves in an extended (3+1)-dimensional Ito equation: state transitions and interactions

Dynamics of transformed nonlinear waves in an extended (3+1)-dimensional Ito equation: state transitions and interactions

The nonlinear dynamics analysis of stochastic delay Jeffcott rotor-seal system with the elastic support

The establishment of stochastic delay Jeffcott rotor-seal system not only considers the track irregularity caused by random noise and the possible influence of stochastic parameter excitation, but also considers the time-delay characteristics of sealing force. The rotation of the rotor shaft causes the gas in the chamber to produce a dynamic effect, resulting in a rotating force, which delays the feedback on the rotating shaft. Firstly, the one-dimensional average Itoˆ differential equation is obtained by simplifying the infinite dimensional stochastic delay differential equation with the perturbation method. Secondly, the global and local stability of the rotor system are obtained by analyzing the singular boundary theory and the maximum Lyapunov exponent. Then, the conditions and types of stochastic bifurcation of rotor system are obtained by analyzing the steady-state joint probability density function. Finally, numerical simulation verifies the accuracy of the theoretical analysis, showing that the time delay affects rotor system to reach the stable critical speed for the first time, and the noise disturbance has a certain stability effect on the rotor system.

Bäcklund transformation, Lax pair and dynamic behaviour of exact solutions for a ($$3+1$$)-dimensional nonlinear equation

Bäcklund transformation, Lax pair and dynamic behaviour of exact solutions for a ($$3+1$$)-dimensional nonlinear equation

Interactions of certain localized waves for an extended (3+1)-dimensional Kadomtsev-Petviashvili equation in fluid mechanics

Currently, there is much interest in the study on the localized waves and their interactions in fluid mechanics. An extended (3+1)-dimensional Kadomtsev-Petviashvili equation is considered here. By means of the Hirota method, we obtain the N-soliton solutions, with N as a positive integer. The higher-order breather solutions are obtained from the N-soliton solutions through the complex conjugated transformations. Making use of the long-wave limit method, we determine the higher-order lump solutions via the N-soliton solutions. Besides, some hybrid solutions are presented. Three kinds of the localized waves, namely, the solitons, breathers and lumps, along with their interactions, are investigated via the above solutions. Amplitudes, shapes and velocities of those localized waves remain invariant after the interactions, which indicate that the interactions are elastic. Fluid-mechanically meaningful, all our results rely on the coefficients in that equation.

The excitation of high-order localized waves in (3+1)-dimensional Kudryashov-Sinelshchikov equation

The aim of this work is to explore the excitation of high-order localized waves in the (3+1)-dimensional Kudryashov-Sinelshchikov equation, which is used to describe the dynamic of liquid with gas bubble. First of all, classical N-soliton solutions are constructed by means of Hirota bilinear form and symbolic calculation. What’s more, the high-order breather waves are derived through the degeneration process of the N-soliton solutions with conjugate parameter. Then, high-order lump waves are constructed by taking long wave limit technique on N-soliton solutions. Finally, the high-order mixed localized waves involving resonant Y-type solitons, high-order breather waves and high-order lump waves are obtained by utilizing some comprehensive methods. Abundant dynamical and evolutionary behaviors of these results are investigated specifically, some figures are presented to shed light on the nonlinear phenomena hidden in the high-order localized waves vividly.

Computational analysis on transient MHD free convection flow in a microchannel in presence of inclined magnetic field with Hall and ion slip current

The present analysis is concerned with the effect of magnetic field inclination on transient MHD flow of Newtonian viscous fluid in a vertical microchannel with the consideration of Hall and ion slip currents as well as induced magnetic field effects. Obtained dimensional partial differential equation are rendered dimensionless by employing suitable parameters and thereafter solved numerically in MATLAB. Relevant actions of parameters on different flow features are depicted explicitly and also using Tables for various applicable parameters. Analysis in this direction is relevant in many MHD controlled applications. Results obtained from the present analysis shows that at the early stages of time and in the simultaneous occurrence of inclined magnetic field as well as Hall and ion slip currents, velocity and induced magnetic field behavior are found to be oscillatory all through the microchannel domain.

Soliton solutions to generalized (3 + 1)-dimensional shallow water-like equation using the (ϕ'/ϕ,1/ϕ)-expansion method

The (3 + 1)-dimensional generalized shallow water equation is a significant mathematical framework for analyzing the dynamic behavior of waves in ocean physics. The purpose of this article is to investigate some more generic soliton solutions of the generalized shallow water-like model in three dimensions. The investigation is conducted utilizing the sophisticated mathematical methodology known as the double variables ( ϕ ′ / ϕ , 1 / ϕ ) expansion technique. With this approach, we produce new propagating wave solutions for this model in the form of hyperbolic, trigonometric, and rational functions. In addition, we offer two- and three-dimensional graphical representations to help visualize the intricate physical phenomena of the system. We have constructed many soliton solutions, such as kink shape soliton solutions, anti-bell shape solutions, single periodic solutions, singular soliton solutions, and anti-kink shape solutions for different values of the free parameters involved in the obtained solutions. These graphical representations are predicated on certain parameter selections, which facilitate the understanding of the complicated general behavior for this model. Through the presentation of new findings in the field of soliton solutions for the aforementioned equation, this paper offers fresh perspectives and highlights hitherto overlooked aspects of this fascinating mathematical challenge. The paper illuminates new results on soliton solutions with different geometrical structures for the given equation, revealing hitherto overlooked facets of this intriguing mathematical challenge.

Three-wave lump solutions and their dynamic behaviors for the (3+1)-dimensional constant-coefficient and variable-coeffcient differential equations

In this paper, we propose two theorems to illustrate the types of equations that can be solved using the quadratic function method to derive the lump solutions localized in the whole plane, which are called three-wave lump solutions, and provide two constant-coefficient equations to illustrate. We further extend the quadratic function method to the variable-coefficient differential equations and obtain the three-wave lump solutions for two (3+1)-dimensional variable-coefficient equations. Moreover, the amplitudes of these lump waves and the distances between the two valleys of each lump are also obtained. Meanwhile, the motion trails, displacements and the velocities of these lump waves are analyzed in detail by virtue of numerical simulation. The study can be used to describe the motion of nonlinear waves in shallow water under the influence of time, and the results can enrich the types of solutions for the KdV-type equations. In addition, the 3d plots and corresponding density plots of the lump waves are displayed to show their spatial structures.

Mixed convection flow of radiated Sutterby nanofluid by stretchable cylinder with irreversibility and heat generation

Entropy generation has gained consideration of researchers due to its applications. Applications of entropy occur in chillers, desert collars, refrigerators, and all type of heat transfer devices. Due to vast range of applications of entropy, in this study irreversibility in Sutterby nanofluid flow by stretchable cylinder is discussed. While reporting momentum equation magnetic field and mixed convection effects are considered. Influences of radiation, Joule heating, and heat source are deliberated in the expression for thermal energy. Chemical reaction impact is taken in the modeling of concentration equation. Irreversibility for the considered flow is obtained by utilizing thermodynamics second law. Dimensional partial differential equations (PDEs) representing the flow are transformed to dimensionless ordinary differential equations(ODEs) through similarity transformations. Solution of ODEs is obtained via NDSolve code of Mathematica. Impact of flow parameters on velocity, entropy, temperature, Bejan number, and concentration are studied graphically. Engineering quantities are analyzed numerically. It is noticed through achieved results that velocity drops for higher magnetic variable while upsurges for higher curvature variable. Thermal field boosts for higher magnetic and heat source parameter. For rising diffusion and radiation variables entropy improves.

Quasi-periodic waves to the defocusing nonlinear Schrödinger equation

A direct approach for the quasi-periodic wave solutions to the defocusing nonlinear Schrödinger equation is proposed based on the theta functions and Hirota’s bilinear method. We transform the problem into a system of algebraic equations, which can be formulated into a least squares problem and then solved by using numerical iterative methods. A rigorous asymptotic analysis demonstrates that these solutions can be classified into two categories: quasi-periodic oscillatory waves and quasi-periodic dark solitons. Singular behaviors may arise in the former case. The numerical results obtained for both the (1+1) -dimensional and (2+1) -dimensional equations are consistent with the theoretical results. Additionaly, the system of algebraic equations can be further extended to address the Riemann–Schottky problem for hyperelliptic curves with 2 infinite points.

Invariant analysis, exact solutions, and conservation laws of time fractional thin liquid film equations

This present paper investigates Lie symmetry analysis, one-dimensional optimal system, exact solutions and conservation laws of the (2 + 1)-dimensional time fractional thin liquid film equations (TFTLFE) with Riemann–Liouville fractional derivative. Explicitly, we obtain six vector fields and the one-dimensional optimal system admitted by TFTLFE. Then, we perform the symmetry reductions with the help of Erdélyi–Kober fractional differential operator and (2 + 1)-dimensional TFTLFE is reduced into (1 + 1)-dimensional fractional partial differential equations (FPDEs). Additionally, by means of compound variable transformation and the power series expansion method, the solution of reduced FPDEs is obtained and its convergence is verified. Moreover, we derive other solutions for the reduced equations taking advantage of the invariant subspace method. Furthermore, the conservation laws are also established utilizing generalized Noether's theorem. Finally, we construct the exact solution using the method of conservation laws.

Optimal Gevrey Regularity for Supercritical Quasi-Geostrophic Equations

We consider the two dimensional surface quasi-geostrophic equations with super-critical dissipation. For large initial data in critical Sobolev and Besov spaces, we prove optimal Gevrey regularity endowed with the same decay exponent as the linear part. This settles several open problems in Biswas (J Differ Equ 257(6):1753–1772, 2014), Biswas et al. (J Funct Anal 269(10):3083–3119, 2015).

Dynamics characteristics of soliton structures of the new (3 + 1) dimensional integrable wave equations with stability analysis

This study investigates optical solitons solutions within the framework of the two new (3 + 1)-dimensional integrable wave equations by employing the modified Sardar sub-equation method. The importance of these suggested models span diverse fields including geophysics, seismology, medical imaging, photonics and sensor development. The derived solutions, meticulously verified using Mathematica, encompass a rich array of mathematical functions including trigonometric, hyperbolic, and exponential functions. Visualization techniques, such as 3D plots, 2D plots, density graphs, contour graphs and polar graphs, effectively illustrate the diverse behaviors exhibited by these soliton solutions. These behaviors include lump, periodic, kink, dark, bright, peakons, cuspons, compactons soliton waves and other complex phenomena. Moreover the modulation instability of the consider model is examined, and corresponding conditions are established. The soliton solutions highlight the efficiency and effectiveness of these method in discovering traveling wave solutions, presenting a valuable tool for addressing a range of nonlinear evolution equations arising in diverse scientific fields such as hydrodynamics, optical fiber communications, plasma physics, ocean engineering, nonlinear dynamics, and condensed matter physics.

Resonance solitons, soliton molecules and hybrid solutions for a ($$\mathbf{2+ 1}$$)-dimensional nonlinear wave equation arising in the shallow water wave

Resonance solitons, soliton molecules and hybrid solutions for a ($$\mathbf{2+ 1}$$)-dimensional nonlinear wave equation arising in the shallow water wave

Soliton Solution of the Nonlinear Time Fractional Equations: Comprehensive Methods to Solve Physical Models

In this paper, we apply two different methods, namely, the G′G-expansion method and the G′G2-expansion method to investigate the nonlinear time fractional Harry Dym equation in the Caputo sense and the symmetric regularized long wave equation in the conformable sense. The mentioned nonlinear partial differential equations (NPDEs) arise in diverse physical applications such as ion sound waves in plasma and waves on shallow water surfaces. There exist multiple wave solutions to many NPDEs and researchers are interested in analytical approaches to obtain these multiple wave solutions. The multi-exp-function method (MEFM) formulates a solution algorithm for calculating multiple wave solutions to NPDEs and at the end of paper, we apply the MEFM for calculating multiple wave solutions to the (2 + 1)-dimensional equation.