Kudryashov Sinelshchikov Equation Research Articles

Kudryashov-Sinelshchikov equation: phase portraits, bifurcation analysis and solitary waves

Kudryashov-Sinelshchikov equation: phase portraits, bifurcation analysis and solitary waves

Multiple interaction solutions, parameter analysis, chaotic phenomena and modulation instability for a (3+1)-dimensional Kudryashov–Sinelshchikov equation in ideal liquid with gas bubbles

Multiple interaction solutions, parameter analysis, chaotic phenomena and modulation instability for a (3+1)-dimensional Kudryashov–Sinelshchikov equation in ideal liquid with gas bubbles

Lax integrability and nonlinear dispersive wave phenomenon for the (3 + 1) dimensional Kudryashov–Sinelshchikov equation

A (3 + 1) dimensional Kudryashov–Sinelshchikov equation is investigated in this paper, which describes bubbles in the liquid fluctuations. By virtue of the binary Bell polynomials, the bilinear representation, bilinear Bäcklund transformation with associated Lax pair are obtained, respectively. Moreover, utilizing Hirota’s bilinear representation, four new lump solutions are constructed and the interaction phenomenon between lump and periodic solution is thoroughly examined. The work also illustrates the intriguing dynamical behavior with the aid of Maple software, which plots the three-dimensional surface, two-dimensional density, and contour profiles of the solutions constructed in this work in various planes.

Optical solitons for the Kudryashov–Sinelshchikov equation by two analytic approaches

Optical solitons for the Kudryashov–Sinelshchikov equation by two analytic approaches

Resonant soliton molecules, asymmetric solitons and the other diverse wave solutions to the (3 + 1)-dimensional generalized Kudryashov-Sinelshchikov equation for liquid with gas bubbles

Under the present study, we focus on developing some exact solutions of the (3 + 1)-dimensional generalized Kudryashov-Sinelshchikov equation (KSE) for the liquid with gas bubbles. First, the resonant soliton molecules on the different planes are extracted via imposing the velocity resonance condition on the N-soliton solutions that constructed via the Hirota bilinear method. Besides, the asymmetric solitons are also derived by choosing the appropriate parameters. In the end, the diverse kinds of travelling wave solutions including the bright wave, dark wave, singular wave and periodic wave solutions that expressed by the trigonometric, hyperbolic and rational functions are constructed by employing three effective approaches, namely the variational method, Sub-equation method and Wang’s direct mapping method-II. The profiles of the attained solutions are depicted to present the dynamical performances. As far as we all know, the results of this research are all new and can enable us figure out the physical properties of the (3 + 1)-dimensional generalized KSE better.

Novel computational technique for fractional Kudryashov–Sinelshchikov equation associated with regularized Hilfer–Prabhakar derivative

Novel computational technique for fractional Kudryashov–Sinelshchikov equation associated with regularized Hilfer–Prabhakar derivative

Solution of the non-linear time-fractional Kudryashov–Sinelshchikov equation using fractional reduced differential transform method

Solution of the non-linear time-fractional Kudryashov–Sinelshchikov equation using fractional reduced differential transform method

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.

Exploring Propagating Soliton Solutions for the Fractional Kudryashov–Sinelshchikov Equation in a Mixture of Liquid–Gas Bubbles under the Consideration of Heat Transfer and Viscosity

In this research work, we investigate the complex structure of soliton in the Fractional Kudryashov–Sinelshchikov Equation (FKSE) using conformable fractional derivatives. Our study involves the development of soliton solutions using the modified Extended Direct Algebraic Method (mEDAM). This approach involves a key variable transformation, which successfully transforms the model into a Nonlinear Ordinary Differential Equation (NODE). Following that, by using a series form solution, the NODE is turned into a system of algebraic equations, allowing us to construct soliton solutions methodically. The FKSE is the governing equation, allowing for heat transmission and viscosity effects while capturing the behaviour of pressure waves in liquid–gas bubble mixtures. The solutions we discover include generalised trigonometric, hyperbolic, and rational functions with kinks, singular kinks, multi-kinks, lumps, shocks, and periodic waves. We depict two-dimensional, three-dimensional, and contour graphs to aid comprehension. These newly created soliton solutions have far-reaching ramifications not just in mathematical physics, but also in a wide range of subjects such as optical fibre research, plasma physics, and a variety of applied sciences.

Dynamics of complexiton, Y-type soliton and interaction solutions to the (3 + 1)-dimensional Kudryashov-Sinelshchikov equation in liquid with gas bubbles

This paper focuses on some novel exact solutions of the (3 + 1)-dimensional Kudryashov-Sinelshchikov equation (KSE). Based on the linear superposition principle (LSP) and weight algorithm (WA), we construct the complexiton solutions by introducing the pairs of the conjugate parameters. Besides, the Y-type soliton solutions are extracted by imposing the resonant conditions on the N-soliton solutions. Finally, two different types of the interaction solutions (ISs) are also explored by exerting the test function method and symbolic computation. The graphical representations of the corresponding solutions are presented to reveal the dynamical characteristics. To the best of our knowledge, the attained outcomes of this article are brand new and have never been reported in other literature, and can enable us make sense of the physical behaviors of the (3 + 1)-dimensional KSE better.

Localized waves and their novel interaction solutions for a dimensionally reduced (2 + 1)-dimensional Kudryashov Sinelshchikov equation

Propagation of the pressure waves in a liquid with gas bubbles is an important topic in the field of fluid dynamics and mathematical physics. The Kudryashov-Sinelshchikov equation is one of the models that describe the propagation of nonlinear waves in a bubbly liquid taking into consideration the viscosity of the liquid and the heat transfer. To explain such behaviors, we mainly focus in this study to explain the dynamics of localized waves and their variety of interaction solutions to a dimensionally reduced (2 + 1)-dimensional Kudryashov-Sinelshchikov equation with the aid of the Hirota bilinear method from N-soliton solutions. Four different forms of localized waves, including solitons, lumps, breathers, and rogues, are derived from the aforesaid equation based on the long wave limit approach. In particular, the localized waves can be used to find interaction solutions, which are the single breather or single lump formed by two solitons; interaction between one line soliton and one breather, as well as one line soliton and one lump soliton among the three solitons; interaction of the two-line soliton and one periodic breather, two periodic breathers, periodic breather and one lump soliton from the four solitons. The direction of propagation, phase shifts, shape, energy, and the variety of interaction solutions of localized waves are affected by these parameters. Moreover, analytical and graphical illustrations of these interaction solutions and their propagation properties are shown by the three-dimensional and density plots with the help of Maple 17. These newly discovered solutions in this study can be used to illustrate the interaction phenomenon of localized waves on ocean surfaces.

Invariant Subspace and Exact Solutions to the Generalized Kudryashov-Sinelshchikov Equation

Invariant Subspace and Exact Solutions to the Generalized Kudryashov-Sinelshchikov Equation

New approximation solution for time-fractional Kudryashov-Sinelshchikov equation using novel technique

In this paper, a novel method presented in [1] is applied to solve time-fractional Kudryashov-Sinelshchikov equation (KS equation), a nonlinear fractional partial differential equation (NFPDE). This method is highly effective in obtaining approximate solutions for strongly NFPDEs. The accuracy of the method is evaluated by estimating the error between the exact and approximate solutions. By applying this method, we obtain solutions for the KS equation at different values of the fractional order derivative and at different stages of time. These solutions are presented through tables and graphs, highlighting the behavior of the KS equation under various conditions.

The generation mechanism of rouge wave for (3 + 1)-dimensional Kudryashov–Sinelshchikov equation

The generation mechanism of rouge wave for (3 + 1)-dimensional Kudryashov–Sinelshchikov equation

Numerical and dynamical behaviors of nonlinear traveling wave solutions of the Kudryashov–Sinelshchikov equation

In this article, a mathematical model symbolizing numerical solutions of the nonlinear Kudryashov–Sinelshchikov (KS) equation has been investigated. For this aim, we use collocation finite element method with septic B-spline functions and appropriate initial and boundary conditions on uniform mesh points. The proficiency and effectiveness of the presented method are examined in four test examples, consisting; behavior of a single soliton, collision of two solitons, Maxwellian initial condition and growth of a wavy hole. L2 and L∞ error norms are used to validate practicality and reliability of our numerical algorithm. In addition, two invariants are calculated to determine the conservation properties of the presented algorithm and the numerical results have been shown to be unconditionally stable. Obtained numerical results have been illustrated with tables and graphics for easy visualization of properties of the problem modeled. Numerical calculations to solve such problems prove the accuracy and effectiveness of the finite element method. In collocation method, using B-spline functions gives rise to a technique that needs only the unknown parameters at certain node points to create the solution. Collocation method has two excellent advantages: establishing method does not include integrations and the resulting matrix system is banded with small band width. Consequently, B-splines associate with the collocation provide a simple solution procedure of linear and nonlinear partial differential equations (PDEs). After that, we have analyzed error estimation for the Galerkin finite element method. Furthermore, dynamical properties of traveling waves of the KS equation are investigated in absence and presence of external periodic perturbation. In absence of perturbation, periodic traveling wave solution of the KS equation are studied. On the other hand, a collection of various quasiperiodic wave features of the KS equation is presented in presence of periodic perturbation. Therefore, it can be said that the numerical technique is effective and valid in obtaining numerical solutions of such PDEs. In addition, the study on dynamical properties of the KS equation provides qualitative behavior of the corresponding wave features.

Approximate analytical solution to the Kudryashov-Sinelshchikov equation with he’s fractional derivative

In this paper, the Adomian decomposition method was employed successfully to solve the Kudryashov-Sinelshchikov equation involving He?s fractional derivatives, and an approximate analytical solution was obtained.

Symmetry Classification and Solutions for the Third-Order of Kudryashov–Sinelshchikov Equation

This paper studies nonlinear wave propagation in a bubble-liquid mixture based on the third-order Kudryashov–Sinelshchikov (KS) equation. Symmetry is classified and reduced through the symmetry group method. Through the generalized conditional symmetry method, this equation is classified, and the generalized conditional symmetry considered are second, third, fourth, and fifth-order, respectively. Meanwhile, the classified equations are transformed into a system of ordinary differential equations and solved.

On the Solutions for a Fifth Order Kudryashov–Sinelshchikov Type Equation

The fifth order Kudryashov–Sinelshchikov equation models the evolution of the nonlinear waves in a gas–liquid mixture, taking into account an interphase heat transfer, surface tension, and weak liquid compressibility simultaneously at the derivation of the equations for non-linear-waves. We prove the well-posedness of the solutions for the Cauchy problem associated with this equation for each choice of the terminal time T.

Pattern Formation of a Bubbly Fluid Mixture under the Effect of Thermodynamics via Kudryashov‐Sinelshchikov Model

In this paper, new explicit wave solutions via liquid‐gas bubbles are obtained for the fractional Kudryashov‐Sinelshchikov (KS) equation under thermodynamic assumptions. A new fractional definition is applied to get these solutions that are utilized to represent the phenomenon of pressure waves under thermodynamic conditions. Two analytical techniques are used to explore the model which is sinh‐Gorden equation expansion and Riccati‐Bernoulli Sub‐ODE methods. These approaches provide complex hyperbolic, hyperbolic, complex trigonometric, and trigonometric solutions for the fractional KS equation, particularly singular, combined singular, dark, bright, combined dark‐bright, and other soliton solutions. Furthermore, acquired results are illustrated by 3D graphs for suitable parametric values that highlight the physical importance and dynamical behaviors of the equation. It is also demonstrated that the purposed approaches are powerful strategies for developing exact traveling wave solutions for a wide range of problems found in mathematical sciences.

Pressure wave characteristics in a bubble-liquid mixture via Kudryashov–Sinelshchikov equation

The general solutions for nonlinear waves in a bubble-liquid mixture are obtained from the Kudryashov–Sinelshchikov (KS) equation under different parametric regimes. The Chiellini integrability condition has been used for constructing exact general solutions. In the non-dissipative case, we find that the solutions may be expressed in terms of Jacobi elliptic and Weierstrass functions. On the other hand, in the dissipative case, only implicit solutions are obtained for the KS equation. For an alternative perspective, the notion of the Jacobi Last Multiplier has been utilized to obtain the corresponding Lagrangian and Hamiltonian description of the reduced equation for the bubble-liquid mixture system.