Search

In this study, the two-dimensional Bogoyavlensky–Konopelchenko equation with variable coefficients (vcBK) which describes the interaction of Riemann waves has been solved two times by using the balance technique. First, the balance procedure was used to reduce the vcBK equation into a fourth-order nonlinear ordinary differential equation and by its implementation again composed with the auxiliary method, many novel periodic, kink and solitary wave solutions were obtained for the vcBK equation. Second, the balance method was used to construct the Bäcklund transformation for the vcBK equation, by which n-soliton solutions were obtained. Finally, some figures were given to show the soliton interaction for different values of the variable coefficients.

In this study, the generalized modified variable-coefficient KdV equation with external-force term (gvcmKdV) describing atmospheric blocking located in the mid-high latitudes over ocean is studied for integrability property by using consistent Riccati expansion solvability and the necessary integrability conditions between the function coefficients are obtained. Moreover, several new solutions have been constructed for the gvcmKdV. Additionally, the classical direct similarity reduction method is used to reduce the gvcmKdV to a nonlinear ordinary differential equation. Building on the solutions given in the previous literature for the reduced equation, many novel solitary and periodic wave solutions have been obtained for the gvcmKdV.

In this study, the two-dimensional Bogoyavlensky–Konopelchenko equation with variable coefficients (vcBK) which describes the interaction of Riemann waves has been solved two times by using the balance technique. First, the balance procedure was used to reduce the vcBK equation into a fourth-order nonlinear ordinary differential equation and by its implementation again composed with the auxiliary method, many novel periodic, kink and solitary wave solutions were obtained for the vcBK equation. Second, the balance method was used to construct the Bäcklund transformation for the vcBK equation, by which n-soliton solutions were obtained. Finally, some figures were given to show the soliton interaction for different values of the variable coefficients. [ABSTRACT FROM AUTHOR]

In this study, the nonautonomous variable coefficients Schrödinger equation describes rogon waves in ocean dynamics and optics, is reduced to the nonlinear ordinary differential equation by using the direct similarity technique. The reduced equation is a Riccati equation of Jacobi elliptic wave type solutions. Therefore, many new Jacobi elliptic wave, periodic and hyperbolic wave solutions are obtained for the nonautonomous variable coefficients Schrödinger equation with some constraints between the variable coefficients. Moreover, a rational solution is given. Finally, many plots for the new rogon wave solutions are investigated. [ABSTRACT FROM AUTHOR]

In this paper, generalized models for both ( 2 + 1 )-dimensional cylindrical modified Korteweg–de Vries (cmKdV) equation with variable coefficients and ( 3 + 1 )-dimensional variable coefficients cylindrical Korteweg–de Vries (cKdV) equation are studied by direct reduction method. A direct reduction to nonlinear ordinary differential equations in the form of Riccati equations obtained for the considered equations under some integrability conditions. The search for solutions for the reduced Riccati equations has yielded many Jacobi elliptic wave solutions, solitary and periodic wave solutions for both ( 2 + 1 )-dimensional cmKdV and ( 3 + 1 )-dimensional cKdV equations. Physical application for the obtained solutions as dust ion acoustic waves in plasma physics is given [ABSTRACT FROM AUTHOR]

In this paper, a generalized variable-coefficient Boiti–Leon–Pempinlli (BLP) system is studied via the modified Clarkson and Kruskal (CK) direct reduction method connected with homogeneous balance (HB) method, which can describe the water waves in fluid physics. A direct similarity reduction to nonlinear ordinary differential system is obtained. By solving the reduced ordinary differential system, new analytical solutions (including solitary and periodic types) in terms of Jacobi elliptic functions are given for the variable-coefficient BLP system. [ABSTRACT FROM AUTHOR]