arXiv subjects

Vijay Kumar Sohani

Publications and source records attributed to Vijay Kumar Sohani.

7 recordsLinked to original sources

arXiv2026

Inhomogeneous nonlinear Schrödinger equation in Fourier-Lebesgue and modulation spaces

The purpose of this work is to provide a broader framework for analyzing the inhomogeneous nonlinear Schrödinger equation (INLS) \[iu_t + u_{xx} \pm |x|^{-b}|u|^{α-1}u=0, \quad 1<α< 5-2b\; \text{and}\; 0< b\leq 1/4.\] 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↗

arXiv2026

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 Schrödinger equation (NLS) having power-type nonlinearity $|u|^{α-1}u$ in a certain modulation spaces $M^{p,p'}(\mathbb{R}),$ where $p'$ is a Hölder 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 $α\in (1,5)$, 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 $1<α\leq 10/3$. 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↗

arXiv2025

On the Boundedness of Hypersingular Integrals Along Certain Radial Hypersurfaces

We study a class of oscillatory hypersingular integral operators associated to a radial hypersurface of the form $Γ(t)=(t,φ(t)), t\in\R{n}$. When $φ$ satisfies suitable curvature and monotonicity conditions, we prove $L^p(\R{n+1})$ boundedness of the operator, where the range of $p$ depends on the hypersingularity of the operator. We also establish certain Sobolev estimates of the operator under consideration.

math.FA↗

arXiv2025

Fractional nonlinear Schrödinger and Hartree equations in modulation spaces

We establish global well-posedness for the mass subcritical nonlinear fractional Schrödinger equation $$iu_t - (-Δ)^\fracβ{2} u+F(u)=0$$ 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 $F(u)=\pm (|u|^αu)\; (0<α<2β/ n)$ or Hartree-type $(|x|^{-ν} \ast |u|^{2})u \; (0<ν<\min\{β,n\}).$ Our order of dispersion $β$ lies in $(2n/ (2n-1), 2).$

math.AP↗

arXiv2024

Low-regularity global solution of the inhomogeneous nonlinear Schrödinger 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 Schrödinger equation (INLS) $$iu_t + Δu\pm |x|^{-b}|u|^αu=0,$$ where $α, b>0,$ on whole space $\mathbb R^n$ in modulation spaces. In the subcritical regime $(0<α< \frac{4-2b}{n}),$ we establish local well-posedness in $L^{2}+M^{α+2,\frac{α+2}{α+1}}( \supset L^2 + H^s \ \text{for} \ s>\frac{nα}{2(α+2)}).$ 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↗

arXiv2013

Nonlinear Schrödinger equation for the twisted Laplacian in the critical case

We prove well-posedness of solution to the nonlinear Schrödinger equation associated to the twisted Laplacian on $\C^n$ for a general class of nonlinearities including power type with subcritical case $0\leq α<\frac{2}{n-1}$, see Ratnakumar, Sohani (J. Funct. Anal. 2013). In this paper, we consider critical case $α=\frac{2}{n-1}$ with $n\geq 2$. Our approach is based on truncation of the given nonlinearity $G$, which is used by Cazenave Weissler (1989). We obtain solution for the truncated problem. We obtain solution to the original problem by passing to the limit.

math.AP↗

arXiv2013

Nonlinear Schrödinger equation for the twisted Laplacian

We establish the local well posedness of solution to the nonlinear Schrödinger equation associated to the twisted Laplacian on $\C^n$ in certain first order Sobolev space. Our approach is based on Strichartz type estimates, and is valid for a general class of nonlinearities including power type. The case $n=1$ represents the magnetic Schrödinger equation in the plane with magnetic potential $A(z)=iz, z\in\C$.

math.AP↗