Rational Localized Waves and Their Absorb-Emit Interactions in the (2 + 1)-Dimensional Hirota–Satsuma–Ito Equation

In this paper, we investigate the (2 + 1)-dimensional Hirota–Satsuma–Ito (HSI) shallow water wave model. By introducing a small perturbation parameter ϵ, an extended (2 + 1)-dimensional HSI equation is derived. Further, based on the Hirota bilinear form and the Hermitian quadratic form, we construct the rational localized wave solution and discuss its dynamical properties. It is shown that the oblique and skew characteristics of rational localized wave motion depend closely on the translation parameter ϵ. Finally, we discuss two different interactions between a rational localized wave and a line soliton through theoretic analysis and numerical simulation: one is an absorb-emit interaction, and the other one is an emit-absorb interaction. The results show that the delay effect between the encountering and parting time of two localized waves leads to two different kinds of interactions.

Lump-Type Wave and Interaction Solutions of the Bogoyavlenskii–Kadomtsev–Petviashvili Equation

Lump-type wave solution of the Bogoyavlenskii–Kadomtsev–Petviashvili equation is constructed by using the bilinear structure and Hermitian quadratic form. The dynamical behaviors of lump-type wave solution are investigated and presented analytically and graphically. Furthermore, we discuss the interaction between a lump-type wave and a kink wave solution. Absorb-emit interaction between two kinds of solitary wave solutions is shown. This kind of interaction solution can be regarded as a lump-type wave which propagates on the kink wave background.

Active Control of Input Power Flow to the Cylindrical Shells Based on Piezoelectric Actuator

The dynamic models of the infinite cylindrical shell with integrated piezoelectric actuator are derived firstly in this paper, then, the total input power flow is calculated and expressed as the Hermitian quadratic form to act as the objective function to implement the control. The optimum set of secondary force is obtained by using feed-forward quadratic optimal theory, and the total input power flow with control was calculated for different locations of the actuator. The results show that different axial and circumferential locations will induce different influences on the control effect, and the results are greatly related to the vibration type and the circumferential mode.

On Weakly Positive Matrices

A matrix is said to be positive definite if it is hermitian and if all of its characteristic values are positive. It is well known, and easy to prove, that the necessary and sufficient condition for a matrix P to be positive definite is that its hermitian quadratic formwith any vector v ≠ 0 be positive. (This will imply, in the present article, that it is real.) It is easy to see from (1) that if P1 and P2 are positive definite, the same holds of a1P1 + a2P2 if a1 and a2 are positive numbers.