arXiv subjects

Diksha Dhingra

Publications and source records attributed to Diksha Dhingra.

4 recordsLinked to original sources

arXiv2026

Inhomogeneous nonlinear Schr\"odinger equation in Fourier-Lebesgue and modulation spaces

The purpose of this work is to provide a broader framework for analyzing the inhomogeneous nonlinear Schr\"odinger equation (INLS) \[iu_t + u_{xx} \pm |x|^{-b}|u|^{\alpha-1}u=0, \quad 1<\alpha< 5-2b\; \text{and}\; 0< b\leq 1/4.\] Specifically, we establish low-regularity well-posedness in the Fourier-Lebesgue $\widehat{L}^{p}$ spaces for $4/3 2$. Primarily, in both cases, we prove global well-posedness for arbitrarily large initial data via the data decomposition method adapted for the Fourier-Lebesgue spaces. Furthermore, we obtain analogous results for INLS in modulation spaces $M^{p,p'}$ for $4/3<p<2$.

math.AP↗

arXiv2025

On 1D mass subcritical nonlinear Schr\''odinger and Hartree equations in modulation spaces $M^{p, p'} \ (p<2)$

We establish well-posedness theory for the 1D mass-subcritical nonlinear Schr\"odinger equation (NLS) having power-type nonlinearity $|u|^{\alpha-1}u$ in a certain modulation spaces $M^{p,p'}(\mathbb{R}),$ where $p'$ is a H\"older conjugate of $p$, with $4/3<p<2$ and $p$ sufficiently close to $2$. Modulation spaces have been successfully applied in understanding the dynamics of NLS near the Sobolev scaling critical regularity. In fact, despite cubic NLS is ill-posed in $H^s$ for $s<-1/2$, our analysis reveals that it experiences well-posedness in modulation spaces for a Cauchy data in $(H^{s} \setminus L^{2}) \cap M^{p,p'}$. The proof adopts two different approaches to establish local well-posedness for $\alpha \in (1,5)$, one exploits generalised Strichartz estimates in Fourier-Lebesgue and Lebesgue spaces; the other implements Bourgain's high-low decomposition (BHLD) method in the modulation space setting. The local solution via the (BHLD) method can be extended to global-in-time, but with a certain loss of regularity. We could combine these effectively and establish global well-posedness in $M^{p,p'}$ with the persistence of regularity for $1<\alpha \leq 10/3$. This is the first global result in $M^{p,p'}$ which establishes the persistence of regularity. Similar results are also established for the Hartree equations.

math.AP↗

arXiv2024

Fractional nonlinear Schr\"odinger and Hartree equations in modulation spaces

We establish global well-posedness for the mass subcritical nonlinear fractional Schr\"odinger equation $$iu_t - (-\Delta)^\frac{\beta}{2} u+F(u)=0$$ having radial initial data in modulation spaces $M^{p,\frac{p}{p-1}}(\mathbb R^n)$ for $n \geq 2, p>2$ and $p$ sufficiently close to $2.$ The nonlinearity $F(u)$ is either of power-type $F(u)=\pm (|u|^{\alpha}u)\; (0<\alpha<2\beta / n)$ or Hartree-type $(|x|^{-\nu} \ast |u|^{2})u \; (0<\nu<\min\{\beta,n\}).$ Our order of dispersion $\beta$ lies in $(2n/ (2n-1), 2).$

math.AP↗

arXiv2024

Low-regularity global solution of the inhomogeneous nonlinear Schr\"odinger equations in modulation spaces

The study of low regularity Cauchy data for nonlinear dispersive PDEs has successfully been achieved using modulation spaces $M^{p,q}$ in recent years. In this paper, we study the inhomogeneous nonlinear Schr\"odinger equation (INLS) $$iu_t + \Delta u\pm |x|^{-b}|u|^{\alpha}u=0,$$ where $\alpha, b>0,$ on whole space $\mathbb R^n$ in modulation spaces. In the subcritical regime $(0<\alpha< \frac{4-2b}{n}),$ we establish local well-posedness in $L^{2}+M^{\alpha+2,\frac{\alpha+2}{\alpha+1}}( \supset L^2 + H^s \ \text{for} \ s>\frac{n\alpha}{2(\alpha+2)}).$ By adapting Bourgain's high-low decomposition method, we establish global well-posedness in $M^{p,\frac{p}{p-1}}$ with $2<p$ and $p$ sufficiently close to 2. This is the first global well-posedness result for INLS on modulation spaces, which contains certain Sobolev $H^s$ $(0<s<1)$ and $L^p_s-$Sobolev spaces.

math.AP↗