Some Non-Linear Evolution Equations and Their Explicit Smooth Solutions with Exponential Growth Written into Integral Form - Petar Popivanov; Angela Slavova (Mathematics, 2024)
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 <semantics>u=be−ax2</semantics>, <semantics>a<0</semantics>, <semantics>a,b</semantics> being real-valued functions. We are looking for the solutions u of Schrödinger-type equation of the form <semantics>u=be−ax22</semantics>, respectively, for the third-order PDE, <semantics>u=AeiΦ</semantics>, where the amplitude b and the phase function a are complex-valued functions, <semantics>A>0</semantics>, and <semantics>Φ</semantics> is real-valued. In our proofs, the method of the first integral is used, not Hirota’s approach or the method of simplest equation.