arXiv subjects

Vishal Vasan

Publications and source records attributed to Vishal Vasan.

15 recordsLinked to original sources

arXiv2026

Growth phases of an active tissue: determinate, indeterminate, and proportionate

Growth may cease at a target size or continue throughout life: the determinate and indeterminate phenotypes. We develop an active viscoelastic continuum model of a tissue growing along one axis, in which cell division and death generate active stresses. We find two asymptotic states: one in which the tissue reaches a relative size fixed by its material parameters, and one in which it elongates linearly without bound. Which state is realised is set by the ratio of active stress to elastic modulus. The transition originates in a bound on the elastic stress the tissue can support: a sufficiently large activity can never be balanced. In a tissue made of parts with different material properties, the growing phase settles into fixed length proportions, set by the mechanical impedances of the parts rather than inherited; matching impedances to initial lengths preserves the proportions the tissue began with. Determinate, indeterminate and proportionate growth thus appear as regimes of one continuum mechanical framework.

physics.bio-ph↗

arXiv2025

Influence of boundary geometry on active patterns

Mechanochemical patterns arising in the actomyosin cortex drive many cellular processes. Here we consider a hydrodynamic model for the actomyosin cortex of cells and study the sensitivity of the emergent patterns to both physical parameters and the geometry of the confining domain. We first establish a general framework for the Galerkin analysis of such patterns far from the linear stability regime on an arbitrary two-dimensional domain. In the case of a circular disk, our analytical results predict transitions from isotropic to anisotropic patterns upon changing the strength of the active stress and the turnover rate. We confirm the existence of these genuine nonlinear bifurcations by an explicit numerical analysis of our model. Extending our numerical analysis to harmonic deformations of the circular disk, we show that the emergent patterns are also sensitive to the curvature of the domain. In particular, the actomyosin patterns resulting from our study closely resemble those seen in cells confined to micropatterned substrates. Our study demonstrates the role of geometry in controlling patterns within the context of a simple model for the actomyosin cortex.

cond-mat.stat-mech↗

arXiv2025

Emergence of coupled Korteweg-de Vries equations in $m$ fields

The Korteweg-de Vries (KdV) equation is of fundamental importance in a wide range of subjects with generalization to multi-component systems relevant for multi-species fluids and cold atomic mixtures. We present a general framework in which a family of multi-component KdV (mKdV) equations naturally arises from a broader mathematical structure under reasonable assumptions on the nature of the nonlinear couplings. In particular, we derive a universal form for such a system of $m$ KdV equations that is parameterized by $m$ non-zero real numbers and two symmetric functions of those $m$ numbers. Secondly, we show that physically relevant setups such as $N\geq m+1$ multi-component nonlinear Schrödinger equations (MNLS), under scaling and perturbative treatment, reduce to such a mKdV equation for a specific choice of the symmetric functions. The reduction from MNLS to mKdV requires one to be in a suitable parameter regime where the associated sound speeds are repeated. Hence, we connect the assumptions made in the derivation of mKdV system to physically interpretable assumptions for the MNLS equation. Lastly, our approach provides a systematic foundation for facilitating a natural emergence of multi-component partial differential equations starting from a general mathematical structure.

math-ph↗

arXiv2023

Explicit Runge-Kutta algorithm to solve non-local equations with memory effects: case of the Maxey-Riley-Gatignol equation

A standard approach to solve ordinary differential equations, when they describe dynamical systems, is to adopt a Runge-Kutta or related scheme. Such schemes, however, are not applicable to the large class of equations which do not constitute dynamical systems. In several physical systems, we encounter integro-differential equations with memory terms where the time derivative of a state variable at a given time depends on all past states of the system. Secondly, there are equations whose solutions do not have well-defined Taylor series expansion. The Maxey-Riley-Gatignol equation, which describes the dynamics of an inertial particle in nonuniform and unsteady flow, displays both challenges. We use it as a test bed to address the questions we raise, but our method may be applied to all equations of this class. We show that the Maxey-Riley-Gatignol equation can be embedded into an extended Markovian system which is constructed by introducing a new dynamical co-evolving state variable that encodes memory of past states. We develop a Runge-Kutta algorithm for the resultant Markovian system. The form of the kernels involved in deriving the Runge-Kutta scheme necessitates the use of an expansion in powers of $t^{1/2}$. Our approach naturally inherits the benefits of standard time-integrators, namely a constant memory storage cost, a linear growth of operational effort with simulation time, and the ability to restart a simulation with the final state as the new initial condition.

physics.flu-dyn↗

arXiv2021

Accurate solution method for the Maxey-Riley equation, and the effects of Basset history

The Maxey-Riley equation has been extensively used by the fluid dynamics community to study the dynamics of small inertial particles in fluid flow. However, most often, the Basset history force in this equation is neglected. Including the Basset force in numerical solutions of particulate flows involves storage requirements which rapidly increase in time. Thus the significance of the Basset history force in the dynamics has not been understood. In this paper, we show that the Maxey-Riley equation in its entirety can be exactly mapped as a forced, time-dependent Robin boundary condition of the one-dimensional heat equation, and solved using the Unified Transform Method. We obtain the exact solution for a general homogeneous time-dependent flow field, and apply it to a range of physically relevant situations. In a particle coming to a halt in a quiescent environment, the Basset history force speeds up the decay as stretched-exponential at short time while slowing it down to a power-law relaxation, $\sim t^{-3/2}$, at long time. A particle settling under gravity is shown to relax even more slowly to its terminal velocity ($\sim t^{-1/2}$), whereas this relaxation would be expected to take place exponentially fast if the history term were to be neglected. For a general flow, our approach makes possible a numerical scheme for arbitrary but smooth flows without increasing memory demands and with spectral accuracy. We use our numerical scheme to solve an example spatially varying flow of inertial particles in the vicinity of a point vortex. We show that the critical radius for caustics formation shrinks slightly due to history effects. Our scheme opens up a method for future studies to include the Basset history term in their calculations to spectral accuracy, without astronomical storage costs. Moreover our results indicate that the Basset history can affect dynamics significantly.

physics.flu-dyn↗

arXiv2021

Ocean-depth measurement using shallow-water wave models

In this paper, we consider a problem inspired by the real-world need to identify the topographical features of ocean basins. Specifically we consider the problem of estimating the bottom impermeable boundary to an inviscid, incompressible, irrotational fluid from measurements of the free-surface deviation alone, within the context of dispersive shallow-water wave models. The need to consider the shallow-water regime arises from the ill-posed nature of the problem and is motivated by prior work. We design an algorithm using which, both fluid velocities and the bottom-boundary profile, may be accurately recovered assuming an a priori relatively inaccurate guess for the bottom boundary. We achieve this by considering two separate inverse problems: one to deduce the bottom-boundary from velocities and the surface deviation, and another to recover the velocities from the surface deviation and an approximate bottom-boundary. The former is a classic inverse problem that requires the inversion of an ill-conditioned matrix while the latter employs the observer framework. Combining the two inverse problems leads to our reconstruction algorithm. We emphasise the role played by model selection and its impact on algorithm design and the accuracy of the reconstruction.

math.AP↗

arXiv2021

Soliton-like behaviour in non-integrable systems

We present a general scheme for constructing robust excitations (soliton-like) in non-integrable multicomponent systems. By robust, we mean localised excitations that propagate with almost constant velocity and which interact cleanly with little to no radiation. We achieve this via a reduction of these complex systems to more familiar effective chiral field-theories using perturbation techniques and the Fredholm alternative. As a specific platform, we consider the generalised multicomponent Nonlinear Schrödinger Equations (MNLS) with arbitrary interaction coefficients. This non-integrable system reduces to uncoupled Korteweg-de Vries (KdV) equations, one for each sound speed of the system. This reduction then enables us to exploit the multi-soliton solutions of the KdV equation which in turn leads to the construction of soliton-like profiles for the original non-integrable system. We demonstrate that this powerful technique leads to the coherent evolution of excitations with minimal radiative loss in arbitrary non-integrable systems. These constructed coherent objects for non-integrable systems bear remarkably close resemblance to true solitons of integrable models. Although we use the ubiquitous MNLS system as a platform, our findings are a major step forward towards constructing excitations in generic continuum non-integrable systems.

nlin.PS↗

arXiv2020

Spatio-temporal relationships between rainfall and convective clouds during Indian Monsoon through a discrete lens

The Indian monsoon, a multi-variable process causing heavy rains during June-September every year, is very heterogeneous in space and time. We study the relationship between rainfall and Outgoing Longwave Radiation (OLR, convective cloud cover) for monsoon between 2004-2010. To identify, classify and visualize spatial patterns of rainfall and OLR we use a discrete and spatio-temporally coherent representation of the data, created using a statistical model based on Markov Random Field. Our approach clusters the days with similar spatial distributions of rainfall and OLR into a small number of spatial patterns. We find that eight daily spatial patterns each in rainfall and OLR, and seven joint patterns of rainfall and OLR, describe over 90\% of all days. Through these patterns, we find that OLR generally has a strong negative correlation with precipitation, but with significant spatial variations. In particular, peninsular India (except west coast) is under significant convective cloud cover over a majority of days but remains rainless. We also find that much of the monsoon rainfall co-occurs with low OLR, but some amount of rainfall in Eastern and North-western India in June occurs on OLR days, presumably from shallow clouds. To study day-to-day variations of both quantities, we identify spatial patterns in the temporal gradients computed from the observations. We find that changes in convective cloud activity across India most commonly occur due to the establishment of a north-south OLR gradient which persists for 1-2 days and shifts the convective cloud cover from light to deep or vice versa. Such changes are also accompanied by changes in the spatial distribution of precipitation. The present work thus provides a highly reduced description of the complex spatial patterns and their day-to-day variations, and could form a useful tool for future simplified descriptions of this process.

physics.ao-ph↗

arXiv2020

Provable bounds for the Korteweg-de Vries reduction in multi-component Nonlinear Schrodinger Equation

We study the dynamics of multi-component Bose gas described by the Vector Nonlinear Schrödinger Equation (VNLS), aka the Vector Gross--Pitaevskii Equation (VGPE) . Through a Madelung transformation, the VNLS can be reduced to coupled hydrodynamic equations in terms of multiple density and velocity fields. Using a multi-scaling and a perturbation method along with the Fredholm alternative, we reduce the problem to a Korteweg de-Vries (KdV) system. This is of great importance to study more transparently, the obscure features hidden in VNLS. This ensures that hydrodynamic effects such as dispersion and nonlinearity are captured at an equal footing. Importantly, before studying the KdV connection, we provide a rigorous analysis of the linear problem. We write down a set of theorems along with proofs and associated corollaries that shine light on the conditions of existence and nature of eigenvalues and eigenvectors of the linear problem. This rigorous analysis is paramount for understanding the nonlinear problem and the KdV connection. We provide strong evidence of agreement between VNLS systems and KdV equations by using soliton solutions as a platform for comparison. Our results are expected to be relevant not only for cold atomic gases, but also for nonlinear optics and other branches where VNLS equations play a defining role.

math-ph↗

arXiv2019

Linear Dispersive Shocks

We present a linear dispersive partial differential equation which manifests a number of qualitative features of dispersive shocks, typically thought to occur only in nonlinear models. The model captures much of the short time phenomenon but deviates from the full nonlinear model in its long time behavior. Though we limit our present discussion to dispersive shocks occurring in the Korteweg-de Vries equation, our work extends readily to other higher order dispersive models too.

math.AP↗

arXiv2018

A Discrete View of the Indian Monsoon to Identify Spatial Patterns of Rainfall

We propose a representation of the Indian summer monsoon rainfall in terms of a probabilistic model based on a Markov Random Field, consisting of discrete state variables representing low and high rainfall at grid-scale and daily rainfall patterns across space and in time. These discrete states are conditioned on observed daily gridded rainfall data from the period 2000-2007. The model gives us a set of 10 spatial patterns of daily monsoon rainfall over India, which are robust over a range of user-chosen parameters as well as coherent in space and time. Each day in the monsoon season is assigned precisely one of the spatial patterns, that approximates the spatial distribution of rainfall on that day. Such approximations are quite accurate for nearly 95% of the days. Remarkably, these patterns are representative (with similar accuracy) of the monsoon seasons from 1901 to 2000 as well. Finally, we compare the proposed model with alternative approaches to extract spatial patterns of rainfall, using empirical orthogonal functions as well as clustering algorithms such as K-means and spectral clustering.

stat.AP↗

arXiv2018

Spatio-temporal Patterns of Indian Monsoon Rainfall

The primary objective of this paper is to analyze a set of canonical spatial patterns that approximate the daily rainfall across the Indian region, as identified in the companion paper where we developed a discrete representation of the Indian summer monsoon rainfall using state variables with spatio-temporal coherence maintained using a Markov Random Field prior. In particular, we use these spatio-temporal patterns to study the variation of rainfall during the monsoon season. Firstly, the ten patterns are divided into three families of patterns distinguished by their total rainfall amount and geographic spread. These families are then used to establish `active' and `break' spells of the Indian monsoon at the all-India level. Subsequently, we characterize the behavior of these patterns in time by estimating probabilities of transition from one pattern to another across days in a season. Patterns tend to be `sticky': the self-transition is the most common. We also identify most commonly occurring sequences of patterns. This leads to a simple seasonal evolution model for the summer monsoon rainfall. The discrete representation introduced in the companion paper also identifies typical temporal rainfall patterns for individual locations. This enables us to determine wet and dry spells at local and regional scales. Lastly, we specify sets of locations that tend to have such spells simultaneously, and thus come up with a new regionalization of the landmass.

stat.AP↗

arXiv2017

Fokas's Uniform Transform Method for linear systems

We demonstrate the use of the Unified Transform Method or Method of Fokas for boundary value problems for systems of constant-coefficient linear partial differential equations. We discuss how the apparent branch singularities typically appearing in the global relation are removable, allowing the method to proceed, in essence, as for scalar problems. We illustrate the use of the method with boundary value problems for the Klein-Gordon equation and the linearized Fitzhugh-Nagumo system. The case of wave equations is treated separately in an appendix.

math.AP↗

arXiv2014

Comparison of five methods of computing the Dirichlet-Neumann operator for the water wave problem

We compare the effectiveness of solving Dirichlet-Neumann problems via the Craig-Sulem (CS) expansion, the Ablowitz-Fokas-Musslimani (AFM) implicit formulation, the dual AFM formulation (AFM*), a boundary integral collocation method (BIM), and the transformed field expansion (TFE) method. The first three methods involve highly ill-conditioned intermediate calculations that we show can be overcome using multiple-precision arithmetic. The latter two methods avoid catastrophic cancellation of digits in intermediate results, and are much better suited to numerical computation. For the Craig-Sulem expansion, we explore the cancellation of terms at each order (up to 150th) for three types of wave profiles, namely band-limited, real-analytic, or smooth. For the AFM and AFM* methods, we present an example in which representing the Dirichlet or Neumann data as a series using the AFM basis functions is impossible, causing the methods to fail. The example involves band-limited wave profiles of arbitrarily small amplitude, with analytic Dirichlet data. We then show how to regularize the AFM and AFM* methods by over-sampling the basis functions and using the singular value decomposition or QR-factorization to orthogonalize them. Two additional examples are used to compare all five methods in the context of water waves, namely a large-amplitude standing wave in deep water, and a pair of interacting traveling waves in finite depth.

math.NA↗

arXiv2011

Recovering the Water-Wave Profile from Pressure Measurements

A new method is proposed to recover the water-wave surface elevation from pressure data obtained at the bottom of the fluid. The new method requires the numerical solution of a nonlocal nonlinear equation relating the pressure and the surface elevation which is obtained from the Euler formulation of the water-wave problem without approximation. From this new equation, a variety of different asymptotic formulas are derived. The nonlocal equation and the asymptotic formulas are compared with both numerical data and physical experiments. The solvability properties of the nonlocal equation are rigorously analyzed using the Implicit Function Theorem.

nlin.PS↗