The simplest parabolic equation of normal type with periodic boundary condition is considered, and the problem of stabilization to zero of its solution with arbitrary initial condition by starting control supported in a prescribed subset is investigated. This problem is reduced to one inequality for starting control, and the proof of this inequality is given. Introduction. This paper is devoted...

This paper studies the generalized version of the ZakharovKuznetsov Benjamin-Bona-Mahoney equation. The functional variable method as well as the simplest equation method are applied to obtain solitons and singular periodic solutions to the equation. There are several constraint conditions that are naturally revealed in order for these specialized type of solutions to exist. The results of this...

In this paper, we study a generalized Zakharov–Kuznetsov equation in three variables, which has applications in the nonlinear development of ion-acoustic waves in a magnetized plasma. Conservation laws for this equation are constructed for the first time by using the new conservation theorem of Ibragimov. Furthermore, new exact solutions are obtained by employing the Lie symmetry method along w...

Abstract—In this paper, the general Riccati equation is analytically solved by a new transformation. By the method developed, looking at the transformed equation, whether or not an explicit solution can be obtained is readily determined. Since the present method does not require a proper solution for the general solution, it is especially suitable for equations whose proper solutions cannot be ...

The basics of the thermodynamic Bethe ansatz equation are given. The simplest case is repulsive delta function bosons, the thermodynamic equation contains only one unknown function. We also treat the XXX model with spin 1/2 and the XXZ model and the XYZ model. This method is very useful for the investigation of the low temperature thermodynamics of solvable systems.

Abstract The extended Kadomtsev-Petviashvili (eKP) equation in fluids is investigated by using Lie symmetry analysis. reductions, together with the simplest method, are used to obtain exact solutions of eKP equation. Finally, we derive conservation laws multiplier approach.