In this article, the magnetohydrodynamic (MHD) thermal boundary layer of a Carreau flow of Cu-water nanofluids over an exponentially permeable stretching thin plate is investigated numerically. Internal heat source/sink is also taken into account. After gaining the system of leading equations, the appropriate transformations have been first employed to come across the fitting parallel conversions to alter the central governing equations into a suit of ODEs and then the renovated system of ODE along with appropriate boundary conditions is numerically solved by the shooting method with fourth-order Runge-Kutta technique. The consequences of the relevant factors of physical parameters on velocity and temperature of merging water (H₂O) and nanoparticles (Cu) have been exemplified through graphs.

In this paper, the cubic-quartic nonlinear Schrödinger and resonant nonlinear Schrödinger equation in parabolic law media are investigated to obtain the dark, singular, bright-singular combo and periodic soliton solutions. Two powerful methods, the $\left(m+\frac{G^{\prime }}{G}\right)$ improved expansion method and the exp $\left(-\phi \left(\xi \right)\right)$ expansion method are utilized to construct some novel solutions of the governing equations. The obtained optical soliton solutions are presented graphically to clarify their physical parameters. Moreover, to verify the existence solutions, the constraint conditions are utilized.

In this paper, we use the modified auxiliary expansion method to seek some new solutions of the paraxial nonlinear Schrödinger equation. The solutions have a hyperbolic function, trigonometric function, exponential function, and rational function forms The linear stability analysis of paraxial NLSE is also presented to study the modulational instability (MI). Two cases when the instability modulation becomes to occur are investigated. Depending on MI cases, the MI gain spectrum are also investigated and presented graphically. All solutions are new and verified the main equation of the paraxial wave equation. Moreover, the constraint conditions for the existence of soliton solutions are also showed.

The multiple Exp-function method is employed for searching the multiple soliton solutions for the new extended (3+1)-dimensional Jimbo-Miwa-like (JM) equation, the extended (2+1)-dimensional Calogero-Bogoyavlenskii-Schiff (eCBS) equation, the generalization of the (2+1)-dimensional Bogoyavlensky-Konopelchenko (BK) equation, and a variable-coefficient extension of the DJKM (vDJKM) equation, which contain one-soliton-, two-soliton-, and triple-soliton-kind solutions. The physical phenomena of these gained multiple soliton solutions are analyzed and indicated in figures by selecting suitable values.