Some Non-Linear Evolution Equations and Their Explicit Smooth Solutions with Exponential Growth Written into Integral Form.

In this paper, exact solutions of semilinear equations having exponential growth in the space variable x are found. Semilinear Schrödinger equation with logarithmic nonlinearity and third-order evolution equations arising in optics with logarithmic and power-logarithmic nonlinearities are investigated. In the parabolic case, the solution u is written as u = b e − a x 2 , a < 0 , a , b being real-valued functions. We are looking for the solutions u of Schrödinger-type equation of the form u = b e − a x 2 2 , respectively, for the third-order PDE, u = A e i Φ , where the amplitude b and the phase function a are complex-valued functions, A > 0 , and Φ is real-valued. In our proofs, the method of the first integral is used, not Hirota's approach or the method of simplest equation. [ABSTRACT FROM AUTHOR]