We consider the Cauchy problem of the non-linear Schrödinger equation with the modulated dispersion and power type non-linearities in any spatial dimensions. We adapt the Young integral theory developed by Chouk–Gubinelli [7] and multilinear estimates which are based on divisor counting and show the local well-posedness. This generalizes the result by Chouk–Gubinelli [7] in terms of the dimension and the order of the non-linearity.

In this article, we investigate the following non-linear Schrödinger (NLS) equation with Neumann boundary conditions:

\begin{equation*}\begin{cases}-\Delta u+ \lambda u= f(u) & {\mathrm{in}} \,~ \Omega,\\\displaystyle\frac{\partial u}{\partial \nu}=0 \, &{\mathrm{on}}\,~\partial \Omega\end{cases}\end{equation*} $$\begin{equation*}\begin{cases}-\Delta u+ \lambda u= f(u) & {\mathrm{in}} \,~ \Omega,\\\displaystyle\frac{\partial u}{\partial \nu}=0 \, &{\mathrm{on}}\,~\partial \Omega\end{cases}\end{equation*}$$

coupled with a constraint condition:

\begin{equation*}\int_{\Omega}|u|^2 dx=c,\end{equation*} $$\begin{equation*}\int_{\Omega}|u|^2 dx=c,\end{equation*}$$

where \Omega\subset \mathbb{R}^N(N\ge3)$\Omega\subset \mathbb{R}^N(N\ge3)$ denotes a smooth bounded domain, ν represents the unit outer normal vector to \partial \Omega$\partial \Omega$, c is a positive constant, and λ acts as a Lagrange multiplier. When the non-linearity f exhibits a general mass supercritical growth at infinity, we establish the existence of normalized solutions, which are not necessarily positive solutions and can be characterized as mountain pass type critical points of the associated constraint functional. Our approach provides a uniform treatment of various non-linearities, including cases such as f(u)=|u|^{p-2}u$f(u)=|u|^{p-2}u$, |u|^{q-2}u+ |u|^{p-2}u$|u|^{q-2}u+ |u|^{p-2}u$, and -|u|^{q-2}u+|u|^{p-2}u$-|u|^{q-2}u+|u|^{p-2}u$, where 2 \lt q \lt 2+\frac{4}{N} \lt p \lt 2^*$2 \lt q \lt 2+\frac{4}{N} \lt p \lt 2^*$. The result is obtained through a combination of a minimax principle with Morse index information for constrained functionals and a novel blow-up analysis for the NLS equation under Neumann boundary conditions.

Recently, we analysed spontaneous symmetry breaking (SSB) of solitons in linearly coupled dual-core waveguides with fractional diffraction and cubic nonlinearity. In a practical context, the system can serve as a model for optical waveguides with the fractional diffraction or Bose–Einstein condensate of particles with Lévy index \alpha <2$\alpha <2$. In an earlier study, the SSB in the fractional coupler was identified as the bifurcation of subcritical type, becoming extremely subcritical in the limit of \alpha \rightarrow 1$\alpha \rightarrow 1$. There, the moving solitons and collisions between them at low speeds were also explored. In the present paper, we present new numerical results for fast solitons demonstrating restoration of symmetry in post-collision dynamics.

We study the existence and strong instability of standing wave solutions for the fractional nonlinear Schrödinger equation

\begin{equation*}\begin{cases}\mathrm{i} \psi_{t}=\left(-\Delta\right)^s \psi-\left(|\psi|^{p-2}\psi+\mu|\psi|^{q-2}\psi\right), & \quad(t,x)\in(0,\infty)\times\mathbb{R}^N,\\\psi(0,x)=\psi_0(x),&\quad x\in\mathbb{R}^N,\end{cases}\end{equation*} $$\begin{equation*}\begin{cases}\mathrm{i} \psi_{t}=\left(-\Delta\right)^s \psi-\left(|\psi|^{p-2}\psi+\mu|\psi|^{q-2}\psi\right), & \quad(t,x)\in(0,\infty)\times\mathbb{R}^N,\\\psi(0,x)=\psi_0(x),&\quad x\in\mathbb{R}^N,\end{cases}\end{equation*}$$

where N\geq2$N\geq2$, 0 \lt s \lt 1$0 \lt s \lt 1$, 2 \lt q \lt p \lt 2_s^*=2N/(N-2s)$2 \lt q \lt p \lt 2_s^*=2N/(N-2s)$, and \mu\in\mathbb{R}$\mu\in\mathbb{R}$. The primary challenge lies in the inhomogeneity of the nonlinearity.We deal with the following three cases: (i) for 2 \lt q \lt p \lt 2+4s/N$2 \lt q \lt p \lt 2+4s/N$ and µ < 0, there exists a threshold mass a0 for the existence of the least energy normalized solution; (ii) for 2+4s/N \lt q \lt p \lt 2_s^*$2+4s/N \lt q \lt p \lt 2_s^*$ and µ > 0, we reveal the existence of the ground state solution, explore the strong instability of standing waves, and provide a blow-up criterion; (iii) for 2 \lt q\leq2+4s/N \lt p \lt 2_s^*$2 \lt q\leq2+4s/N \lt p \lt 2_s^*$ and µ < 0, the strong instability of standing wave solutions is demonstrated. These findings are illuminated through variational characterizations, the profile decomposition, and the virial estimate.

The article studies an initial boundary valueproblem (ibvp) for the radial solutions of the nonlinear Schrödinger (NLS) equation in a radially symmetric region \Omega\in \mathbb R^n$\Omega\in \mathbb R^n$ with boundaries. All such regions can be classified into three types: a ball Ω0 centred at origin, a region Ω1 outside a ball, and an n-dimensional annulus Ω2. To study the well-posedness of those ibvps, the function spaces for the boundary data must be specified in terms of the solutions in appropriate Sobolev spaces. It is shown that when \Omega = \Omega_1$\Omega = \Omega_1$, the ibvp for the NLS equation is locally well-posed in C( [0, T^*]; H^s(\Omega_1))$C( [0, T^*]; H^s(\Omega_1))$ if the initial data is in H^s(\Omega_1)$H^s(\Omega_1)$ and boundary data is in H^{\frac{2s+1}{4}}(0, T)$H^{\frac{2s+1}{4}}(0, T)$ with s \geq 0$s \geq 0$. This is the optimal regularity for the boundary data and cannot be improved. When \Omega = \Omega_2$\Omega = \Omega_2$, the ibvp is locally well-posed in C( [0, T^*]; H^s(\Omega_2))$C( [0, T^*]; H^s(\Omega_2))$ if the initial data is in H^s(\Omega_2)$H^s(\Omega_2)$ and boundary data is in H^{\frac{s+1}{2}}(0, T)$H^{\frac{s+1}{2}}(0, T)$ with s \geq 0$s \geq 0$. In this case, the boundary data requires 1/4$1/4$ more derivative compared to the case when \Omega = \Omega_1$\Omega = \Omega_1$. When \Omega = \Omega_0$\Omega = \Omega_0$ with n = 2 (the case with n > 2 can be discussed similarly), the ibvp is locally well-posed in C( [0, T^*]; H^s(\Omega_0))$C( [0, T^*]; H^s(\Omega_0))$ if the initial data is in H^s(\Omega_0)$H^s(\Omega_0)$ and boundary data is in H^{\frac{s+1}{2}}(0, T)$H^{\frac{s+1}{2}}(0, T)$ with s > 1 (or s \gt n/2$s \gt n/2$). Due to the lack of Strichartz estimates for the corresponding boundary integral operator with 0 \leq s \leq 1$0 \leq s \leq 1$, the local well-posedness can only be achieved for s > 1. It is noted that the well-posedness results on Ω0 and Ω2 are the first ones for the ibvp of NLS equations in bounded regions of higher dimension.

In this paper, we consider blowup of solutions to the Cauchy problem for the following biharmonic nonlinear Schrödinger equation (NLS),

\begin{equation*}\text{i } \partial_t u=\Delta^2 u-\mu \Delta u-|u|^{2 \sigma} u \quad \text{in} \,\, \mathbb{R} \times \mathbb{R}^d,\end{equation*} $$\begin{equation*}\text{i } \partial_t u=\Delta^2 u-\mu \Delta u-|u|^{2 \sigma} u \quad \text{in} \,\, \mathbb{R} \times \mathbb{R}^d,\end{equation*}$$

where d \geq 1$d \geq 1$, \mu \in \mathbb{R}$\mu \in \mathbb{R}$ and 0 \lt \sigma \lt \infty$0 \lt \sigma \lt \infty$ if 1 \leq d \leq 4$1 \leq d \leq 4$ and 0 \lt \sigma \lt 4/(d-4)$0 \lt \sigma \lt 4/(d-4)$ if d \geq 5$d \geq 5$. In the mass critical and supercritical cases, we establish the existence of blowup solutions to the problem for cylindrically symmetric data. The result extends the known ones with respect to blowup of solutions to the problem for radially symmetric data.

We study Gibbs measures with log-correlated base Gaussian fields on the d-dimensional torus. In the defocusing case, the construction of such Gibbs measures follows from Nelson’s argument. In this paper, we consider the focusing case with a quartic interaction. Using the variational formulation, we prove nonnormalizability of the Gibbs measure. When d = 2$d = 2$, our argument provides an alternative proof of the nonnormalizability result for the focusing \Phi ^4_2$\Phi ^4_2$-measure by Brydges and Slade (1996). Furthermore, we provide a precise rate of divergence, where the constant is characterized by the optimal constant for a certain Bernstein’s inequality on \mathbb R^d$\mathbb R^d$. We also go over the construction of the focusing Gibbs measure with a cubic interaction. In the appendices, we present (a) nonnormalizability of the Gibbs measure for the two-dimensional Zakharov system and (b) the construction of focusing quartic Gibbs measures with smoother base Gaussian measures, showing a critical nature of the log-correlated Gibbs measure with a focusing quartic interaction.

We prove that the cubic nonlinear Schrödinger equation (both focusing and defocusing) is globally well-posed in H^s({{\mathbb {R}}})$H^s({{\mathbb {R}}})$ for any regularity s>-\frac 12$s>-\frac 12$. Well-posedness has long been known for s\geq 0$s\geq 0$, see [55], but not previously for any s<0$s<0$. The scaling-critical value s=-\frac 12$s=-\frac 12$ is necessarily excluded here, since instantaneous norm inflation is known to occur [11, 40, 48].

We also prove (in a parallel fashion) well-posedness of the real- and complex-valued modified Korteweg–de Vries equations in H^s({{\mathbb {R}}})$H^s({{\mathbb {R}}})$ for any s>-\frac 12$s>-\frac 12$. The best regularity achieved previously was s\geq \tfrac 14$s\geq \tfrac 14$ (see [15, 24, 33, 39]).

To overcome the failure of uniform continuity of the data-to-solution map, we employ the method of commuting flows introduced in [37]. In stark contrast with our arguments in [37], an essential ingredient in this paper is the demonstration of a local smoothing effect for both equations. Despite the nonperturbative nature of the well-posedness, the gain of derivatives matches that of the underlying linear equation. To compensate for the local nature of the smoothing estimates, we also demonstrate tightness of orbits. The proofs of both local smoothing and tightness rely on our discovery of a new one-parameter family of coercive microscopic conservation laws that remain meaningful at this low regularity.

In this paper, we are concerned with the ground states of the following planar Kirchhoff-type problem:

-\left(1+b\displaystyle\int_{\mathbb{R}^2}|\nabla u|^2\,{\rm d}x\right)\Delta u+\omega u=|u|^{p-2}u, \quad x\in\mathbb{R}^2. $$-\left(1+b\displaystyle\int_{\mathbb{R}^2}|\nabla u|^2\,{\rm d}x\right)\Delta u+\omega u=|u|^{p-2}u, \quad x\in\mathbb{R}^2.$$

b,\, \omega >0$b,\, \omega >0$

p>2$p>2$

We consider the two-dimensional minimisation problem for \inf \{ E_a(\varphi ):\varphi \in H^1(\mathbb {R}^2)\ \text {and}\ \|\varphi \|_2^2=1\}$\inf \{ E_a(\varphi ):\varphi \in H^1(\mathbb {R}^2)\ \text {and}\ \|\varphi \|_2^2=1\}$, where the energy functional E_a(\varphi )$E_a(\varphi )$ is a cubic-quintic Schrödinger functional defined by E_a(\varphi ):=\tfrac 12\int _{\mathbb {R}^2}|\nabla \varphi |^2\,dx-\tfrac 14a\int _{\mathbb {R}^2}|\varphi |^4\,dx+\tfrac 16a^2\int _{\mathbb {R}^2}|\varphi |^6\,dx$E_a(\varphi ):=\tfrac 12\int _{\mathbb {R}^2}|\nabla \varphi |^2\,dx-\tfrac 14a\int _{\mathbb {R}^2}|\varphi |^4\,dx+\tfrac 16a^2\int _{\mathbb {R}^2}|\varphi |^6\,dx$. We study the existence and asymptotic behaviour of the ground state. The ground state \varphi _{a}$\varphi _{a}$ exists if and only if the L^2$L^2$ mass a satisfies a>a_*={\lVert Q\rVert }^2_2$a>a_*={\lVert Q\rVert }^2_2$, where Q is the unique positive radial solution of -\Delta u+ u-u^3=0$-\Delta u+ u-u^3=0$ in \mathbb {R}^2$\mathbb {R}^2$. We show the optimal vanishing rate \int _{\mathbb {R}^2}|\nabla \varphi _{a}|^2\,dx\sim (a-a_*)$\int _{\mathbb {R}^2}|\nabla \varphi _{a}|^2\,dx\sim (a-a_*)$ as a\searrow a_*$a\searrow a_*$ and obtain the limit profile.

The Schrödinger–Poisson system describes standing waves for the nonlinear Schrödinger equation interacting with the electrostatic field. In this paper, we are concerned with the existence of positive ground states to the planar Schrödinger–Poisson system with a nonlinearity having either a subcritical or a critical exponential growth in the sense of Trudinger–Moser. A feature of this paper is that neither the finite steep potential nor the reaction satisfies any symmetry or periodicity hypotheses. The analysis developed in this paper seems to be the first attempt in the study of planar Schrödinger–Poisson systems with lack of symmetry.

We consider the following Kirchhoff-type problem in an unbounded exterior domain \Omega\subset\mathbb{R}^{3}$\Omega\subset\mathbb{R}^{3}$:(*)

\begin{align}\left\{\begin{array}{ll}-\left(a+b\displaystyle{\int}_{\Omega}|\nabla u|^{2}\,{\rm d}x\right)\triangle u+\lambda u=f(u), & x\in\Omega,\\\\u=0, & x\in\partial \Omega,\\\end{array}\right.\end{align} $$\begin{align}\left\{\begin{array}{ll}-\left(a+b\displaystyle{\int}_{\Omega}|\nabla u|^{2}\,{\rm d}x\right)\triangle u+\lambda u=f(u), & x\in\Omega,\\\\u=0, & x\in\partial \Omega,\\\end{array}\right.\end{align}$$

b\geq0$b\geq0$

\partial\Omega\neq\emptyset$\partial\Omega\neq\emptyset$

\mathbb{R}^{3}\backslash\Omega$\mathbb{R}^{3}\backslash\Omega$

u\in H_{0}^{1}(\Omega)$u\in H_{0}^{1}(\Omega)$

f\in C^1(\mathbb{R},\mathbb{R})$f\in C^1(\mathbb{R},\mathbb{R})$

\begin{equation*}-\Delta u+\lambda u=f(u), \qquad x\in \mathbb R^3 \end{equation*} $$\begin{equation*}-\Delta u+\lambda u=f(u), \qquad x\in \mathbb R^3 \end{equation*}$$

H^1(\mathbb R^3)$H^1(\mathbb R^3)$

\mathbb R^3\setminus \Omega$\mathbb R^3\setminus \Omega$

We consider the following class of quasilinear Schrödinger equations proposed in plasma physics and nonlinear optics -\Delta u+V(x)u+\frac {\kappa }{2}[\Delta (u^{2})]u=h(u)$-\Delta u+V(x)u+\frac {\kappa }{2}[\Delta (u^{2})]u=h(u)$ in the whole two-dimensional Euclidean space. We establish the existence and qualitative properties of standing wave solutions for a broader class of nonlinear terms h(s)$h(s)$ with the critical exponential growth. We apply the dual approach to obtain solutions in the usual Sobolev space H^{1}(\mathbb {R}^{2})$H^{1}(\mathbb {R}^{2})$ when the parameter \kappa >0$\kappa >0$ is sufficiently small. Minimax techniques, Trudinger–Moser inequality and the Nash–Moser iteration method play an essential role in establishing our results.

We introduce a numerical framework for dispersive equations embedding their underlying resonance structure into the discretisation. This will allow us to resolve the nonlinear oscillations of the partial differential equation (PDE) and to approximate with high-order accuracy a large class of equations under lower regularity assumptions than classical techniques require. The key idea to control the nonlinear frequency interactions in the system up to arbitrary high order thereby lies in a tailored decorated tree formalism. Our algebraic structures are close to the ones developed for singular stochastic PDEs (SPDEs) with regularity structures. We adapt them to the context of dispersive PDEs by using a novel class of decorations which encode the dominant frequencies. The structure proposed in this article is new and gives a variant of the Butcher–Connes–Kreimer Hopf algebra on decorated trees. We observe a similar Birkhoff type factorisation as in SPDEs and perturbative quantum field theory. This factorisation allows us to single out oscillations and to optimise the local error by mapping it to the particular regularity of the solution. This use of the Birkhoff factorisation seems new in comparison to the literature. The field of singular SPDEs took advantage of numerical methods and renormalisation in perturbative quantum field theory by extending their structures via the adjunction of decorations and Taylor expansions. Now, through this work, numerical analysis is taking advantage of these extended structures and provides a new perspective on them.

We consider the \mathbb {T}^{4}$\mathbb {T}^{4}$ cubic nonlinear Schrödinger equation (NLS), which is energy-critical. We study the unconditional uniqueness of solutions to the NLS via the cubic Gross–Pitaevskii hierarchy, an uncommon method for NLS analysis which is being explored [24, 35] and does not require the existence of a solution in Strichartz-type spaces. We prove U-V multilinear estimates to replace the previously used Sobolev multilinear estimates. To incorporate the weaker estimates, we work out new combinatorics from scratch and compute, for the first time, the time integration limits, in the recombined Duhamel–Born expansion. The new combinatorics and the U-V estimates then seamlessly conclude the H^{1}$H^{1}$ unconditional uniqueness for the NLS under the infinite-hierarchy framework. This work establishes a unified scheme to prove H^{1}$H^{1}$ uniqueness for the \mathbb {R}^{3}/\mathbb {R}^{4}/\mathbb {T}^{3}/\mathbb {T}^{4}$\mathbb {R}^{3}/\mathbb {R}^{4}/\mathbb {T}^{3}/\mathbb {T}^{4}$ energy-critical Gross–Pitaevskii hierarchies and thus the corresponding NLS.

The aim of the paper is to explore non-local reverse-space matrix non-linear Schrödinger equations and their inverse scattering transforms. Riemann–Hilbert problems are formulated to analyse the inverse scattering problems, and the Sokhotski–Plemelj formula is used to determine Gelfand–Levitan–Marchenko-type integral equations for generalised matrix Jost solutions. Soliton solutions are constructed through the reflectionless transforms associated with poles of the Riemann–Hilbert problems.

A fundamental question in wave turbulence theory is to understand how the wave kinetic equation describes the long-time dynamics of its associated nonlinear dispersive equation. Formal derivations in the physics literature, dating back to the work of Peierls in 1928, suggest that such a kinetic description should hold (for well-prepared random data) at a large kinetic time scale T_{\mathrm {kin}} \gg 1$T_{\mathrm {kin}} \gg 1$ and in a limiting regime where the size L of the domain goes to infinity and the strength \alpha $\alpha$ of the nonlinearity goes to 0$0$ (weak nonlinearity). For the cubic nonlinear Schrödinger equation, T_{\mathrm {kin}}=O\left (\alpha ^{-2}\right )$T_{\mathrm {kin}}=O\left (\alpha ^{-2}\right )$ and \alpha $\alpha$ is related to the conserved mass \lambda $\lambda$ of the solution via \alpha =\lambda ^2 L^{-d}$\alpha =\lambda ^2 L^{-d}$ .

In this paper, we study the rigorous justification of this monumental statement and show that the answer seems to depend on the particular scaling law in which the (\alpha , L)$(\alpha , L)$ limit is taken, in a spirit similar to how the Boltzmann–Grad scaling law is imposed in the derivation of Boltzmann’s equation. In particular, there appear to be two favourable scaling laws: when \alpha $\alpha$ approaches 0$0$ like L^{-\varepsilon +}$L^{-\varepsilon +}$ or like L^{-1-\frac {\varepsilon }{2}+}$L^{-1-\frac {\varepsilon }{2}+}$ (for arbitrary small \varepsilon $\varepsilon$ ), we exhibit the wave kinetic equation up to time scales O(T_{\mathrm {kin}}L^{-\varepsilon })$O(T_{\mathrm {kin}}L^{-\varepsilon })$ , by showing that the relevant Feynman-diagram expansions converge absolutely (as a sum over paired trees). For the other scaling laws, we justify the onset of the kinetic description at time scales T_*\ll T_{\mathrm {kin}}$T_*\ll T_{\mathrm {kin}}$ and identify specific interactions that become very large for times beyond T_*$T_*$ . In particular, the relevant tree expansion diverges absolutely there. In light of those interactions, extending the kinetic description beyond T_*$T_*$ toward T_{\mathrm {kin}}$T_{\mathrm {kin}}$ for such scaling laws seems to require new methods and ideas.

In this paper, we focus on a Kirchhoff-type equation with singular potential and critical exponent. By virtue of the generalized version of Lions-type theorem and the Nehari–Pohožaev manifold, we established the existence of Nehari–Pohožaev-type ground state solutions to the mentioned equation. Some recent results from the literature are generally improved and extended.

We construct global-in-time singular dynamics for the (renormalized) cubic fourth-order nonlinear Schrödinger equation on the circle, having the white noise measure as an invariant measure. For this purpose, we introduce the ‘random-resonant / nonlinear decomposition’, which allows us to single out the singular component of the solution. Unlike the classical McKean, Bourgain, Da Prato-Debussche type argument, this singular component is nonlinear, consisting of arbitrarily high powers of the random initial data. We also employ a random gauge transform, leading to random Fourier restriction norm spaces. For this problem, a contraction argument does not work, and we instead establish the convergence of smooth approximating solutions by studying the partially iterated Duhamel formulation under the random gauge transform. We reduce the crucial nonlinear estimates to boundedness properties of certain random multilinear functionals of the white noise.

We consider the focussing fractional periodic Korteweg–deVries (fKdV) and fractional periodic non-linear Schrödinger equations (fNLS) equations, with L2 sub-critical dispersion. In particular, this covers the case of the periodic KdV and Benjamin-Ono models. We construct two parameter family of bell-shaped travelling waves for KdV (standing waves for NLS), which are constrained minimizers of the Hamiltonian. We show in particular that for each \lambda > 0$\lambda > 0$, there is a travelling wave solution to fKdV and fNLS \phi : \|\phi \|_{L^2[-T,T]}^2=\lambda $\phi : \|\phi \|_{L^2[-T,T]}^2=\lambda$, which is non-degenerate. We also show that the waves are spectrally stable and orbitally stable, provided the Cauchy problem is locally well-posed in Hα/2[ − T, T] and a natural technical condition. This is done rigorously, without any a priori assumptions on the smoothness of the waves or the Lagrange multipliers.