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The focus of this article is to find some new exact solutions to the (2+1)-dimensional nonlinear Schrödinger equation (NLSE) with Kerr media by employing the auxiliary equation method. A set of novel travelling wave solutions are observed such as bright, dark, periodic and optical solitons (dromions). Moreover, the physical interpretation of nonlinear waves would also be demonstrated with the aid of scientific computing. [ABSTRACT FROM AUTHOR]

In this work, we retrieve a series of new wave solutions of the double dispersive equation that describes the nonlinear wave propagation in the elastic inhomogeneous Murnaghan's rod by applying a novel integration norm known as the Sardar sub-equation method (SSEM). The solutions recovered by this method can be categorized as topological, non-topological, combo topological–non-topological, periodic and mixed periodic, singular and mixed singular solutions. In addition, by allotting different values to parameters, the physical movements of attained solutions are shown by plotting the 3D, 2D and contour graphs. On the basis of achieved results, we may claim that the proposed computational method is direct, dynamics, well organized, and will be useful for solving the more complicated nonlinear problems in diverse areas together with symbolic computations, especially in engineering and applied sciences. [ABSTRACT FROM AUTHOR]

This paper studies the exact traveling wave solutions to the nonlinear Dullin–Gottwald–Holm model which has the application in shallow-water waves in which the fractional derivative is considered in the sense of conformable derivative. Diverse exact solutions in hyperbolic, trigonometric and plane wave forms are obtained using two integration norms. For this purpose G ′ / G -expansion method and reccati mapping techniques are used. The 3D plots and their corresponding contour graphs are also depicted. Being concise and straightforward, the calculations demonstrate the effectiveness and convenience of the method for solving other nonlinear partial differential equations. [ABSTRACT FROM AUTHOR]

Ternary alloys of Fe are very important materials having good corrosion resistance and are also famous for several high-temperature applications. The dynamical behavior of exact traveling waves for the kinetics of phase separation in Fe–Cr–X (X = Mo, Cu) ternary alloys is modelled by convective-diffusive Cahn–Hilliard (CH) equation. A variety of nonlinear dynamical exact and solitary wave structures are extracted in several forms like rational, hyperbolic, trigonometric function solutions by the utilization of a sound computational integration tool, i.e., Φ 6 -model expansion method. Besides, we also secure mixed combined solitons and singular periodic wave solutions with unknown parameters. Moreover, the constraint conditions observed during derivation lead to substantial solutions. The findings elucidate that the governing model theoretically possesses significantly rich structures of exact traveling wave solutions. Hence, our technique via fortification of symbolic computations provides an active and potent mathematical implementation for solving diverse benevolent nonlinear wave problems. [ABSTRACT FROM AUTHOR]

A particular attention is paid to the nonlinear dynamical exact wave solutions to the (2 + 1)-dimensional Date–Jimbo–Kashiwara–Miwa equation (DJKME). A variety of solutions are extracted in different shapes like dark, singular, dark-singular by implementing ( G ′ G 2 ) -expansion function method and modified direct algebraic method. In addition, we also secure singular periodic and plane wave solutions with arbitrary parameters. We also discussed the modulation instability analysis of the governing model. The constraint conditions for the validity of existence of solutions are also reported. Moreover, three-dimensional and two-dimensional, and their corresponding contour graphs are sketched for a better understanding of the derived solutions with the values of arbitrary parameters. [ABSTRACT FROM AUTHOR]

The focus of this article is to find some new exact solutions to the M-fractional paraxial nonlinear Schrödinger equation with Kerr media by employing the modified simple equation method and the auxiliary equation method. A set of novel travelling wave solutions are observed such as bright, dark, periodic and optical solitons. Moreover, the physical interpretation of nonlinear waves would also be demonstrated with the aid of scientific computing. [ABSTRACT FROM AUTHOR]

This paper studies the new closed form solutions to (2+1)-dimensional Bogoyavlenskii system that describes interaction of a Riemann wave propagation. The extended Fan sub-equation technique is used to investigate some new traveling wave solutions to the higher-dimensional coupled model. The obtained closed form solutions are named as shock, kink, shock and periodic soliton solutions. Clearly, the outcomes of the study confirm the strength of the current approach. Moreover, the obtained results are helpful for the understanding of non-linear wave propagation and are of great interest to present-day scientists and can be employed to deal with more complex models arising in diverse disciplines of contemporary science. [ABSTRACT FROM AUTHOR]