Narrow Results

An extension of the KPII equation is studied. Adding a new fourth-order derivative term and some second-order derivative terms, we formulate an extended KPII equation. Different types of solutions of the extended equation are obtained by the Hirota bilinear method, and the presented solutions include soliton solutions, lump solutions and interaction solutions. Their dynamical behaviors are analyzed through plots.

In this paper, the solitary solutions of the (1 + 1)-dimensional Benjamin–Bona–Mahony and strain wave equations are set up by employing the simplest equation method for the first time. By determining the value of parameters, the images of the traveling wave solutions are displayed, and they intuitively reflect that choosing different Bernoulli equations will affect the form of solutions. The validity and wider applicability of the method for solving known wave phenomena models are verified, moreover, the solution structure is enriched.

This paper uses the Hausdorff fractal derivative to convert the new Schrödinger nonlinear coupled equation into a novel fractal coupled nonlinear Schrödinger model. By applying the variational principle, a plethora of new soliton solutions are retrieved from the developed framework. The conditions of constraints are set for the presence of appropriate solitons. The 3D, 2D, and contour graphs of the reported solutions are depicted under the collection of appropriate parameter values. Moreover, it is noted that the variational principle built on the Hausdorff derivative for the proposed fractal model delivers a direct convenient and efficient mathematical tool for solving nonlinear partial differential equations in the solitary wave theory.

Based on the bifurcation theory, the exact solutions of fractional Date–Jimbo–Kashiwara–Miwa equation are constructed. In terms of $\beta$-derivative, a nonlinear integer-order ODE is obtained, the equation is transformed into a plane dynamical system. The phase portraits are obtained under different bifurcation conditions. According to the orbits of the phase portraits, new bright and dark solitary solutions, periodic solutions, periodic-singular solutions, and singular solutions are established, enriching the diversity of solutions. In addition, the dynamical behaviors of various types of solutions are shown by selecting appropriate parameters, which is useful for understanding the fluctuation behavior in physical models. Compared with previous literature, this paper focuses on the combination of qualitative analyses and qualitative calculations, which makes the paper more systematic and comprehensive, and fills the gap of using bifurcation theory to solve the FDJKM equation. The dynamical behavior of new solutions obtained will provide more and more attractive and complex physical phenomena to explore more mysteries.