Variational properties of space-periodic standing waves of nonlinear Schrödinger equations with general nonlinearities (Q6621522)

The authors consider the 1D nonlinear Schrödinger equation: \(i\psi _{t}+\psi _{xx}+bf(\psi )=0\), where \(\psi :\mathbb{R}_{t}\times \mathbb{R} _{x}\rightarrow \mathbb{C}\), \(f\) is a gauge invariant nonlinearity and \(b\in \mathbb{R}\setminus \{0\}\). They define the associated energy functional: \(\mathcal{E}(\psi )=\frac{1}{2}\int_{0}^{T}\left\vert \psi _{x}\right\vert ^{2}dx-b\int_{0}^{T}F(\psi )dx\), with \(F(z)=\int_{0}^{\left \vert z\right\vert }f(s)ds\) and \(T\) the space period, the mass functional: \( M(\psi )=\frac{1}{2}\int_{0}^{T}\left\vert \psi \right\vert ^{2}dx\), the momentum functional: \(P(\psi )=\frac{1}{2} Im \int_{0}^{T}\psi \overline{\psi _{x}}dx\), the action functional: \(S(\psi )=\mathcal{E}(\psi )-aM(\psi )\), for a given \(a\), and the Nehari functional: \(I(\psi )=\left\langle S^{\prime }(\psi ),\psi \right\rangle =\int_{0}^{T}\left\vert \psi _{x}\right\vert ^{2}dx-a\int_{0}^{T}\left\vert \psi \right\vert ^{2}dx-b\int_{0}^{T} Re(f(\psi )\overline{\psi })dx\). The main result proves properties of these functionals:\ If \(b\) is positive, there exists a real valued minimizer of the energy, under fixed mass or under fixed mass and zero momentum, among periodic functions. The minimal energy is finite and negative. If the mass is larger than a given threshold, then the minimizer is not a constant, the associated Lagrange multiplier verifies \(a<0 \), and the minimizer is positive. If \(b\) is negative, there exists a unique (up to phase shift) minimizer of the energy which is the constant function \( u_{\infty }=\sqrt{2m/T}\), under fixed mass or under fixed mass and zero momentum, among periodic functions. If \(b\) is negative and \( f(u)=\sum_{j=1}^{N}\left\vert u\right\vert ^{p_{j-1}}u\), with \(p_{j}>1\) for \( j=1,\ldots ,N\), there exists a unique (up to phase shift and complex conjugate) minimizer which is the plane wave \(u_{\infty }=\sqrt{2m/T}e^{i\pi x/T}\) of the energy, under fixed mass, among anti-periodic functions. If \(b\) is positive, \(a\) negative and \(f(u)=\left\vert u\right\vert ^{p-1}u\), with \( p>1\), the minimum of the action on the Nehari manifold among periodic functions is finite and there exists a real minimizer. If \(b\) is negative, \(a \) positive and \(f(u)=\left\vert u\right\vert ^{p-1}u\), with \(p>1\), there exists a unique (up to phase shift) minimizer which is the constant function \(u_{\infty }=-(a/b)^{1/(p-1)}\) of the action on the Nehari manifold among periodic functions. If \(b\) is positive, \(a<4\pi ^{2}/T^{2}\) and \( f(u)=\left\vert u\right\vert ^{p-1}u\), with \(p>1\), the minimum of the minimization problem on the Nehari manifold among anti-periodic functions is finite and there exists a minimizer. When \(p\) is an odd integer, this minimizer is real. For the proof, the authors observe that the standing waves are solutions of the form \(\psi (t,x)=e^{-iatu(x)}\), \(a\in \mathbb{R}\), where the profile function \(u(x)\) satisfies the ordinary differential equation \(u_{xx}+au+bf(u)=0\). They analyze the bounded solutions to this profile equation, assuming regularity and growth properties on \(f\). They prove an inequality based on a Fourier rearrangement process. Assuming further hypotheses on the nonlinearity \(f\), they prove that, for all \(m>0\), the minimization problem: \(\min\{\mathcal{E}(u):u\in H_{\mathrm{loc}}^{1}(\mathbb{R})\cap P_{T}\), \(M(u)=m\}\), where \(P_{T}=\{f\in L_{\mathrm{loc}}^{2}(\mathbb{R}):f(x+T)=f(x)\}\), admits a real minimizer which is also a minimizer of the minimization problem: \(\min\{\mathcal{E}(u):u\in H_{\mathrm{loc}}^{1}(\mathbb{R})\cap P_{T}\), \(M(u)=m\), \(P(u)=0\}\). The minimal energy is finite and negative. The authors also prove that if \(m>\widetilde{m}\), for some positive threshold \(\widetilde{m}\), this minimizer is not a constant, that the associated Lagrange multiplier \(a\) is negative, and that the minimizer is positive. They finally prove focusing or defocusing properties on \(P_{T}\) and on \(A_{T}=\{f\in L_{\mathrm{loc}}^{2}(\mathbb{R}):f(x+T)=-f(x)\}\).