Propagation of dust-acoustic nonlinear waves in a homogeneous collisional dusty plasma

Nonlinear propagation of dust-acoustic waves DAWs in a weakly collisional dusty plasma comprising warm adiabatic fluid dust particles, isothermal electrons, and ions is investigated. We used the reductive perturbation theory to reduce the basic set of fluid equations to one evolution equation, namely damped Kadomtsev--Petviashivili (DKP). The analytical stationary solutions of the DKP equation are numerically analyzed, and the effect of various dusty plasma parameters on DAWs wave propagation is taken into account. We obtained, blast, anti-kink, periodic cnoidal and cnoidal waves. It is well known that explosive waves are a double edged sword. They can be seen, for example, in the atmosphere, or in engineering applications in metal coating. _______________________________________________

New Traveling Wave Solutions and Interesting Bifurcation Phenomena of Generalized KdV-mKdV-Like Equation

Using the bifurcation method of dynamical systems, we investigate the nonlinear waves and their limit properties for the generalized KdV-mKdV-like equation. We obtain the following results: (i) three types of new explicit expressions of nonlinear waves are obtained. (ii) Under different parameter conditions, we point out these expressions represent different waves, such as the solitary waves, the 1-blow-up waves, and the 2-blow-up waves. (iii) We revealed a kind of new interesting bifurcation phenomenon. The phenomenon is that the 1-blow-up waves can be bifurcated from 2-blow-up waves. Also, we gain other interesting bifurcation phenomena. We also show that our expressions include existing results.

Spatial Analyticity of Solutions to Korteweg–de Vries Type Equations

The Korteweg–de Vries equation (KdV) is a mathematical model of waves on shallow water surfaces. It is given as third-order nonlinear partial differential equation and plays a very important role in the theory of nonlinear waves. It was obtained by Boussinesq in 1877, and a detailed analysis was performed by Korteweg and de Vries in 1895. In this article, by using multi-linear estimates in Bourgain type spaces, we prove the local well-posedness of the initial value problem associated with the Korteweg–de Vries equations. The solution is established online for analytic initial data w0 that can be extended as holomorphic functions in a strip around the x-axis. A procedure for constructing a global solution is proposed, which improves upon earlier results.

Bifurcations of Nonlinear Waves in the Generalized KdV–mKdV-Like Equation

Using bifurcation analytic method of dynamical systems, we investigate the nonlinear waves and their bifurcations of the generalized KdV–mKdV-like equation. We obtain the following results : (i) Three types of new explicit expressions of nonlinear waves are obtained. They are trigonometric expressions, exp-function expressions, and hyperbolic expressions. (ii) Under different parameteric conditions, these expressions represent different waves, such as solitary waves, kink waves, 1-blow-up waves, 2-blow-up waves, smooth periodic waves and periodic blow-up waves. (iii) Two kinds of new interesting bifurcation phenomena are revealed. The first phenomenon is that the single-sided periodic blow-up waves can bifurcate from double-sided periodic blow-up waves. The second phenomenon is that the double-sided 1-blow-up waves can bifurcate from 2-blow-up waves. Furthermore, we show that the new expressions encompass many existing results.