Comment on “New types of exact solutions for nonlinear Schrodinger equation with cubic nonlinearity”

In this comment we analyze the paper [Abdelhalim Ebaid, S.M. Khaled, New types of exact solutions for nonlinear Schrodinger equation with cubic nonlinearity, J. Comput. Appl. Math. 235 (2011) 1984-1992]. Using the traveling wave, Ebaid and Khaled have found “new types of exact solutions for nonlinear Schrodinger equation with cubic nonlinearity”. We demonstrate that the authors studied the well-known nonlinear ordinary differential equation with the well-known general solution. We illustrate that Ebaid and Khaled have looked for some exact solution for the reduction of the nonlinear Schrodinger equation taking the general solution of the same equation into account.     

The Cauchy problem for the equation of the Burgers hierarchy

The Cauchy problem for the equation of the Burgers hierarchy is considered. The Green function for the associated linear problem is constructed. Using the Cole–Hopf transformation the solution of the Cauchy problem for the equation of the Burgers hierarchy is given. Several particular cases are considered and discussed.     

Exact solutions of the Swift-Hohenberg equation with dispersion

The Swift-Hohenberg equation with dispersion is considered. Traveling wave solutions of the Swift-Hohenberg equation with dispersion are presented. The classification of these solutions is given. It is shown that the Swift-Hohenberg equation without dispersion has only stationary meromorphic solution.     

A note on “New kink-shaped solutions and periodic wave solutions for the (2 + 1)-dimensional Sine-Gordon equation”

Exact solutions of the Nizhnik-Novikov-Veselov equation by Li [New kink-shaped solutions and periodic wave solutions for the (2 + 1)-dimensional Sine-Gordon equation, Appl. Math. Comput. 215 (2009) 3777-3781] are analyzed. We have observed that fourteen solutions by Li from 30 do not satisfy the equation. The other 16 exact solutions by Li can be found from the general solutions of the well-known solution of the equation for the Weierstrass elliptic function.     

Nonlinear Dynamics of a Bubble Contrast Agent Oscillating near an Elastic Wall

Contrast agent microbubbles, which are encapsulated gas bubbles, are widely used to enhance ultrasound imaging. There are also several new promising applications of the contrast agents such as targeted drug delivery and noninvasive therapy. Here we study three models of the microbubble dynamics: a nonencapsulated bubble oscillating close to an elastic wall, a simple coated bubble and a coated bubble near an elastic wall.We demonstrate that complex dynamics can occur in these models. We are particularly interested in the multistability phenomenon of bubble dynamics. We show that coexisting attractors appear in all of these models, but for higher acoustic pressures for the models of an encapsulated bubble.We demonstrate how several tools can be used to localize the coexisting attractors. We provide some considerations why the multistability can be undesirable for applications.     

Integrable Nonautonomous Liénard-Type Equations

We study a family of nonautonomous generalized Liénard-type equations. We consider the equivalence problem via the generalized Sundman transformations between this family of equations and type-I Painlevé–Gambier equations. As a result, we find four criteria of equivalence, which give four integrable families of Liénard-type equations. We demonstrate that these criteria can be used to construct general traveling-wave and stationary solutions of certain classes of diffusion–convection equations. We also illustrate our results with several other examples of integrable nonautonomous Liénard-type equations.     

A Note on the Lie Symmetry Analysis and Exact Solutions for the Extended mKdV Equation

We discuss the recent paper by Liu and Li (Lie symmetry analysis and exact solutions for the extended mKdV equation, Acta Appl. Math. (2010), 109:1107?C1119) and illustrate that so called the ??extended mKdV equation?? by authors can be reduced to the usual form of the modified Korteweg?Cde Vries equation. We also correct some other statements by authors with respect to exact solutions of solutions for the ??extended mKdV equation??.     

Experimental investigation of the formation of double phosphates in the LaPO4–NaF system

In the course of segregation smelting of rare-earth and rare metal raw materials with a fluxing agent (NaF), two immiscible melts form, one of which is a silicate melt and the other is a phosphate–salt melt. The silicate melt is enriched with Fe₂O₃, Al₂O₃, SiO₂, and Nb₂O₅, and the phosphate–salt melt is dominated by P₂O₅, TR₂O₃, Sc₂O₃, and Y₂O₃, and also with Ca, Sr, and Ba oxides. Chemical reactions between lanthanum orthophosphate and sodium fluoride in the LaPO₄–NaF system were studied for developing a technology for processing the phosphate–salt (rare-earth metal) melt. It was found that a metathesis reaction gives double phosphate Na₃La[PO₄]₂ and binary fluoride NaLaF₄. The products of crystallization of melts in the LaPO₄–NaF system decompose in weak mineral acids unlike those in conventional technology for processing monazite raw material.     

Synchronous oscillations and symmetry breaking in a model of two interacting ultrasound contrast agents

Nonlinear Dynamics - We study nonlinear dynamics in a system of two coupled oscillators, describing the motion of two interacting microbubble contrast agents. In the case of identical bubbles, the...