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This work investigates the time–space fractional coupled nonlinear Schrödinger equation. By applying an appropriate wave transformation, this equation is converted into a fourth-order system of ordinary differential equations, equivalent to a Hamiltonian system with two degrees of freedom. The integrability of the Hamiltonian system is examined using Painlevé analysis. We demonstrate that the Hamiltonian system is completely integrable in the Liouville sense in two cases, wherein the Hamilton–Jacobi equation is also separable, and we introduce the non-integrability conditions. The first integrals of motion for the separable cases are provided. Utilizing bifurcation analysis, we depict the phase portrait and subsequently integrate the separable Hamiltonian system and construct new solutions. Some of solutions are illustrated graphically, and it is shown that the wave solutions profiles are sensitive to temporal and spatial fractional derivatives.

This work strives to study the impact of both the multiplicative Wiener process and spatial fractional derivatives on the solutions of (3+1) stochastic fractional partial differential equation describing the fluids with gas bubbles. Based on the complete discriminate polynomial system, we introduce a new extension for the auxiliary function method that is more effective for constructing analytical solutions for stochastic fractional partial differential equations and all special cases resulting from it such as deterministic partial differential equations with integer or fractional order derivatives. An algorithm summarizing our method is inserted to avoid confusion about its applicability. The proposed method has multiple advantages such as it furnishes only real solutions and isolates the bounded and unbounded solutions. This method is successfully applied to (3+1) stochastic fractional partial differential equation describing the fluids with gas bubbles and resulting in some new solutions. Some of these solutions are clarified graphically as well as the individual and combined influences of fractional order and noise on some of the obtained solutions. These effects are some changes in the amplitude and the width of the solutions and the smoothness of the surface represented by these solutions.

We are interested in studying some qualitative analysis and wave propagation and their applications to the Konopelchenko-Dubrovsky equation.Using the qualitative theory of planar systems and a complete discriminant system, we propose a new method. The proposed method is more efficient because it only creates real bounded wave solutions, which are desirable in real-world applications, as well as it is applicable to large classes of nonlinear partial differential equations. We present an algorithm for the proposed method to facilitate and clarify its applicability. Some new solutions are introduced using this method, and they are classified as periodic and solitary wave solutions. These solutions are graphically represented by displaying 3D and 2D graphic representations, as well as the contour plot. Furthermore, we investigate the impact of the included parameters on the obtained solutions. We numerically examine the 2D and 3D phase portraits after permitting a certain periodic additional term to arise, revealing the existence of quasi-periodic behavior.

Our study focuses on the governing equation of a finitely deformed flexible rod with strain waves. By utilizing the well-known Ablowita–Ramani–Segur (ARS) algorithm, we prove that the equation is non-integrable in the Painlevé sense. Based on the bifurcation theory for planar dynamical systems, we modify an auxiliary equation method to obtain a new systematic and effective method that can be used for a wide class of non-linear evolution equations. This method is summed up in an algorithm that explains and clarifies the ease of its applicability. The proposed method is successfully applied to construct wave solutions. The developed solutions are grouped as periodic, solitary, super periodic, kink, and unbounded solutions. A graphic representation of these solutions is presented using a 3D representation and a 2D representation, as well as a 2D contour plot.

This work aims to explore some dynamic aspects of the problem of star motion that is impacted by the rotation of the galaxy, which we model as a bisymmetric potential based on a two-dimensional harmonic oscillator with sextic perturbations. We demonstrate analytically that the motion is non-integrable when certain conditions are met. The analytical results for the non-integrability are confirmed by showing the irregularity of the behavior of the motion through utilizing the Poincaré surface of a section as a numerical method. The motion equilibrium positions are detected, and their stability is discussed. We show that the force generated by the rotating frame acts as a stabilizer for the maximum equilibrium points. We display graphically that the size of the stability regions relies on the angular velocity magnitude for the frame. Through the application of Lyapunov’s theorem, periodic solutions can be constructed which are close to the equilibrium positions. Furthermore, we demonstrate that there are one or two families of periodic solutions relying on whether the equilibrium point is a saddle or stable, respectively.

This paper studies the influence of space-fractional and multiplicative noise on the exact solutions of the space-fractional stochastic dispersive modified Benjamin–Bona–Mahony equation, driven in Ito’s sense by a multiplicative Wiener process. The bifurcation of the exact solutions is investigated, and novel fractional stochastic solutions are presented. The dependence of the solutions on the initial conditions is discussed. Due to the significance of the fractional stochastic modified Benjamin–Bona–Mahony equation in describing the propagation of surface long waves in nonlinear dispersive media, the derived solutions are significantly more helpful for and influential in comprehending diverse, crucial, and challenging physical phenomena. The effect of the Wiener process and the fractional order on the exact solutions are studied.

This paper explores the effects of spatial fractional derivatives and the multiplicative Wiener process on the analytical solutions for (3+1)-dimensional fractional stochastic equations for fluids with gas bubbles. We study the bifurcation of the analytical solutions and introduce new fractional stochastic solutions. We also discuss how the solutions differ depending on the initial conditions. The new solutions are notably more beneficial and impactful for understanding various, significant, and incredibly hard physical phenomena due to the significance of the modified fractional stochastic (3+1)-dimensional equations for fluids with gas bubbles and hydrodynamics. We also discuss the effects of the fractional order and the Wiener process on the obtained analytical solutions.