Collisionless damping of soliton solutions of Korteweg - de Vries equations, Korteweg - de Vries modified equation and nonlinear Schrodinger equation

Background. Solution of nonlinear differential equations in partial derivatives is a topical and complex problem. As distinct from linear differential equations, for which the solution methods have been developed (for example, the Fourier method, the Laplace method etc.), for nonlinear differential equations in partial derivatives there are no general solution methods. Each nonlinear equation or a small group of similar equations require the development of individual specific solution methods. Materials and methods. The article considers nonstationary, damped soliton solutions of three equations (Korteweg - de Vries, Korteweg - de Vries modified equation and nonlinear Schrodinger equation), describing, in particular, various oscillation modes in plasma. Using the method of scale transformations, the authors obtained nonstationary (damped) solutions of the said equations, valid for the case when as a result of interaction of a static soliton ensemble with plasma on functions of electron and (or) ions distribution there is formed a non-Maxwellian high-energy part (“power tale”). Results and conclusions. The obtained solution of the Korteweg - de Vries equation may be applied for magnetosonic plasma waves, propagating at an angle to a magnetic field, the solution of the Korteweg - de Vries modified equation may be applied, for example, in warm dust plasma, containing two types of ions, and the solution of the nonlinear Schrodinger equation is valid, for example, in plasma corona target of laser thermonuclear synthesis near a critical density.