Lump-periodic, some interaction phenomena and breather wave solutions to the (2+1)-rth dispersionless Dym equation

In this study, we successfully apply Hirota’s bilinear method (HBM) to retrieve the different wave structures of the general [Formula: see text]th dispersionless Dym equation by considering the test function approaches. The studied model is used to describe the dynamics of deep water waves. We formally retrieve some novel lump periodic, some other new interaction, and breather wave solutions. Moreover, the physical behavior of the reported results is sketched through several three-dimensional, two-dimensional and contour profiles with the assistance of suitable parameters. The acquired results are valuable in grasping the elementary scenarios of nonlinear fluid dynamics as well as the dynamics of engineering sciences in the related nonlinear higher-dimensional wave fields. The gained results are checked and found correct by putting them into the governing equation with the aid of Mathematica. Thus, our strategies through the fortress of representative calculations give a functioning and intense mathematical execution for tackling complicated nonlinear wave issues.

Integration of the Harry Dym equation with an integral type source

In the work, we deduce the evolution of scattering data for a spectral problem associated with the nonlinear evolutionary equation of Harry Dym with a self-consistent source of integral type. The obtained equalities completely determine the scattering data for any $t$, which makes it possible to apply the method of the inverse scattering problem to solve the Cauchy problem for the Harry Dym equation with an integral type source.

Similarity Solutions to Nonlinear Diffusion/Harry Dym Fractional Equations

By using scalar similarity transformation, nonlinear model of time-fractional diffusion/Harry Dym equation is transformed to corresponding ordinary fractional differential equations, from which a travelling-wave similarity solution of time-fractional Harry Dym equation is presented. Furthermore, numerical solutions of time-fractional diffusion equation are discussed. Again, through another similarity transformation, nonlinear model of space-fractional diffusion/Harry Dym equation is turned into corresponding ordinary differential equations, whose two similarity solutions are also worked out.

Self-Adjointness and Conservation Laws of Frobenius Type Equations

The Frobenius KDV equation and the Frobenius KP equation are introduced, and the Frobenius Kompaneets equation, Frobenius Burgers equation and Frobenius Harry Dym equation are constructed by taking values in a commutative subalgebra Z2ε in the paper. The five equations are selected as examples to help us study the self-adjointness of Frobenius type equations, and we show that the first two equations are quasi self-adjoint and the last three equations are nonlinear self-adjointness. It follows that we give the symmetries of the Frobenius KDV and the Frobenius KP equation in order to construct the corresponding conservation laws.

Dynamics and exact solutions of the generalized Harry Dym equation

The Harry Dym equation is the third-order evolutionary partial differential equation. It describes a system in which dispersion and nonlinearity are coupled together. It is a completely integrable nonlinear evolution equation that may be solved by means of the inverse scattering transform. It has an infinite number of conservation laws and does not have the Painleve property. The Harry Dym equation has strong links to the Korteweg – de Vries equation and it also has many properties of soliton solutions. A connection was established between this equation and the hierarchies of the Kadomtsev – Petviashvili equation. The Harry Dym equation has applications in acoustics: with its help, finite-gap densities of the acoustic operator are constructed. The paper considers a generalization of the Harry Dym equation, for the study of which the methods of the theory of finite-dimensional dynamics are applied. The theory of finite-dimensional dynamics is a natural development of the theory of dynamical systems. Dynamics make it possible to find families that depends on a finite number of parameters among all solutions of evolutionary differential equations. In our case, this approach allows us to obtain some classes of exact solutions of the generalized equation, and also indicates a method for numerically constructing solutions.