Defocusing nonlinear Schrödinger equation

Writing a defocusing nonlinear Schrödinger field as $\psi =\sqrt{\rho }{e}^{iS}$ turns it into a continuity equation for $\rho$ and a velocity equation for $v=S_x$, including a quantum-pressure term.

In a common normalization, the defocusing case is $i{\psi }_t+{\psi }_{xx}-2|\psi {|}^2\psi =0$. The sign distinguishes it from the focusing nonlinear Schrödinger equation.

For the Robin boundary condition $q_x(0,t)=cq(0,t)$ with real $c$, the time Lax pair has a diagonal-conjugation symmetry under $k\mapsto -k$. It implies $A(-k)=A(k)$ and $B(-k)=-h(k)B(k)$. Together with the half-line NLS global relation, this eliminates unknown boundary scattering from an equivalent Riemann-Hilbert problem determined by the initial data and $c$.

Here $\Gamma =B^{♯}/(ad)$ and $\Delta =\mathcal{D}/(2k+ic)$. The difference follows algebraically from the half-line NLS global relation and the Robin symmetry. Multiplication by the spatial-temporal jump factor gives $e^{2ikx-4i{k}^2(T-t)}$, decaying in the second quadrant for $t<T$. A triangular analytic change of the Riemann-Hilbert problem removes this finite-horizon defect while retaining the reconstruction and transferring all genuine pole information to the known determinant. No unproved large-time decay of the boundary values is needed.

The effective boundary coefficient is $\tilde{\Gamma }=-(2k-ic)b^{♯}(-k)/(a\mathcal{D})$. All its ingredients come from initial scattering and the Robin boundary condition. Genuine zeros produce pole data; common zeros in numerator and denominator can instead be removable. It is incorrect to assert absence of boundary poles for every real Robin parameter.

Initial scattering gives $s$ and boundary-time scattering gives $S$. The defocusing reduction and unit determinants give the displayed matrix forms. The initial functions $a,b$ are analytic in the upper half-plane, while the boundary functions $A,B$ are bounded in the first and third quadrants. Their finite-horizon compatibility is the half-line NLS global relation.

Evaluating the Lax pair connection formula at $(0,T)$ gives $S^{-1}s=e^{2i{k}^{2}T{\hat{\sigma }}_3}{\mu }_3(0,T,k)$. The upper off-diagonal entry is the displayed relation, where $c_T=[{\mu }_3(0,T){]}_{12}$ is analytic in the upper half-plane and is $O(1/k)$. A finite-time relation must retain this final-state term.

For $\psi =e^{-iEt}f(x)$ with real $f$, the defocusing equation $i{\psi }_t+{\psi }_{xx}-2|\psi {|}^2\psi =0$ has first integralA nonzero smooth function tending to zero at both infinities cannot satisfy this identity: a nonzero extremum requires $f^2=E>0$, but then the right side is negative for sufficiently small nonzero $f$.

For the normalizationa dark soliton of speed $0\le U<1/\sqrt{2}$ is