In MATLAB solving Nonlinear Nonlocal Schrodinger equation when the nonlinear term is spatially nonlocal

One of the methods solving it is, say,Pseudo-spectral method:

L=80; N=2048; dt=0.0001; tmax=20; nmax=round(tmax/dt);
dx=L/N; x=[-L/2:dx:L/2-dx]'; k=[0:N/2-1 -N/2:-1]'*2*pi/L; k2=k.^2;
u=1.2*sech(1.2*(x+20)).*exp(i*x)+0.8*sech(0.8*x);
udata=u; tdata=0;
for nn=1:nmax                               % integration begins
du1=1i*(ifft(-k2.*fft(u))+2*u.*u.*conj(u));  v=u+0.5*du1*dt;
du2=1i*(ifft(-k2.*fft(v))+2*v.*v.*conj(v));  v=u+0.5*du2*dt;
du3=1i*(ifft(-k2.*fft(v))+2*v.*v.*conj(v));  v=u+    du3*dt;
du4=1i*(ifft(-k2.*fft(v))+2*v.*v.*conj(v));
u=u+(du1+2*du2+2*du3+du4)*dt/6;
if mod(nn,round(nmax/25)) == 0
 udata=[udata u]; tdata=[tdata nn*dt];
end
end                                         % integration ends
mesh(x, tdata, abs(udata'));                % solution plotting
colormap('jet'); view(10, 60)
grid on
text(-2,  -6, 'x', 'fontsize', 15)
text(50, 5, 't', 'fontsize', 15)
zlabel('|u|', 'fontsize', 15)
axis([-L/2 L/2 0 tmax 0 2]); grid off
set(gca, 'xtick', [-40 -20 0 20 40])
set(gca, 'ytick', [0 10 20])
set(gca, 'ztick', [0 1 2])

This code solves the NLSE (STANDARD). In MATLAB the nonlinear term in standard NLSE reads as

u(x,t).*conj(u(x,t)).*u(x,t).

Now in nonlocal NLSE the same nonlinear term is :

u(x,t).*conj(u(-x,t)).*u(x,t)

with a '-' sign in conj(u(x,t)).

Now my question is: In NONLOCAL NLSE how the code is going to be changed, more specifically how the '-' sign of the nonlinear term is going to be taken care of in line no 6 to 9.

Thanks and regards: Samit

Answer 1

[Hafid yacine]

let me see