In this work, we prove the persistence in time of superoscillations for the Schrödinger equation with time-dependent coefficients. In order to prove the persistence of superoscillations, we have conditioned the coefficients to satisfy a Riccati system, and we have expressed the solution as a convolution operator in terms of solutions of this Riccati system. Further, we have solved explicitly the Cauchy initial value problem with three different kinds of superoscillatory initial data. The operator is defined on a space of entire functions. Particular examples include Caldirola-Kanai and degenerate parametric harmonic oscillator Hamiltonians, and other examples could include Hamiltonians not self-adjoint. For these examples, we have illustrated numerically the convergence on real and imaginary parts. KEYWORDS closed and approximate solutions to the Schrödinger equation, evolution of superoscillations, Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type, operators on function spaces, superoscillating functions MSC CLASSIFICATION 81Q05; 47B38; 42A38n (x, 0) = (cos(x∕n) + ia sin(x∕n)) n *Aharonov is also well-known by the Aharonov-Bohm effect.

In this work we study a stochastic susceptible-infected (SI) model with space-uniform white noise. We present conditions to find exact solutions and numerical simulations are compared to these exact solutions in some cases. We demonstrate that the model is exponentially asymptotically stable almost surely. We show the global stability and local stability of a special case of the model.

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We will introduce exact and numerical solutions to some stochastic Burgers equations with variable coefficients. The solutions are found using a coupled system of deterministic Burgers equations and stochastic differential equations.