# Categorize and solve a certain partial differential equation

I derived some partial differential equation.

$$V(x)-u_{xx}-u_x^2=i u_t,$$ where $$u=u(x,t)$$.

I do not know even if this differential equation has some special name. I also do not know how to solve it.

I am looking for numerical solution for the following initial condition $$u(x,0)=-(x-x_0)^2$$ and potential $$V(x)=x^2$$ in range $$x \in [-8,8]$$, $$t \in [0,t_{max}]$$.

• Look into NDSolve – NonDairyNeutrino Sep 21 '18 at 22:21
• I know Mathematica functions. Problem is that simple direct solution does not work. Well, it requires specification of boundary conditions which I do not know. – QuantumNik Sep 21 '18 at 22:24

## 1 Answer

Consider the equation

$$-if_t-f_{xx}+V(x)f=0$$

We make a substitution $$f=e^{u(x,t)}$$.

f = Exp[u[x, t]];
FullSimplify[D[f, t]/I - D[f, x, x] + f*V[x]]

E^u[x, t](V[x] - I Derivative[0, 1][u][x, t] - Derivative[1, 0][u][x, t]^2 - Derivative[2, 0][u][x, t]) Thus, we have obtained an equation that the author considers. But the original equation is the Schrödinger equation. Let us consider a problem with initial data and with boundary conditions. Using the NDSolve we find a numerical solution

L = 4; t0 = 10; x0 = 1;
F0[x_] := Exp[-(x - x0)^2]
sol =
NDSolveValue[
{D[F[x, t], t]/I - D[F[x, t], x, x] + F[x, t]*x^2 == 0,
F[x, 0] == F0[x], F[L, t] == F0[L], F[-L, t] == F0[-L]},
F, {x, -L, L}, {t, 0, t0}]

Plot3D[Abs[sol[x, t]], {x, -L, L}, {t, 0, t0},
Mesh -> None, ColorFunction -> Hue] Phase of the wave function in the plane (x,t). In this case we set L = 8

DensityPlot[Arg[sol[x, t]], {x, -L, L}, {t, 0, t0}, Mesh -> None,
ColorFunction -> Hue, PlotRange -> All, PlotPoints -> 100,
PlotLegends -> Automatic] • Thanks for your comment. The problem is that I deliberately consider the presented equation which was indeed obtained by substitution of $e^u$ into the Schrodinger equation. My initial problem is to obtain the phase of the wavefunction over whole $x$ domain. Try to extract the $u$ function from your calculation. You will see that due to numerical errors it is possible only in the region where the wave packet is large enough. I was trying to avoid this limitation by transforming the SE into the presented one. Thus I am looking for a way to solve the presented equation in the original form. – QuantumNik Sep 24 '18 at 11:13
• @QuantumNik I published a picture with the phase of the wave function. Please specify a problem. – Alex Trounev Sep 24 '18 at 13:32
• Yes, it works in certain situations. Now increase your $x$ domain, so $L=40$. You will see that the phase of the solution will be just a noise outside of a small region near the maximum of the wave packet. Of course, you can increase precision but this is a dead end definitely. I am looking for a method which will allow me to deal with the phase in a different way. Thus I tried to represent it in a form of the exponential of a function. The original problem is to obtain the phase from the solution of the Schrodinger equation. – QuantumNik Sep 24 '18 at 15:15
• @QuantumNik We will not improve this situation with a numerical solution when solving a nonlinear equation. It will only get worse. – Alex Trounev Sep 24 '18 at 16:33