Integrable Systems, Nonlinear Waves & All That

#integrablesystems #solitons #nonlinearwaves #symmetry #symmetries #mathematicalphysics #integrability #PDEs #dynamicalsystems #dynsys #OPSF #ODEs #geometry

[Integrable Systems, Nonlinear Waves & All That @integrablesys]

@academath cc @adhara_mathphys @TamasGorbe @vm_braindumps

[Academe, Math & More @academath]

👉 Did you know that 2005 @abel_prize winner Peter Lax, known for his work both in pure and applied mathematics, was born 100 years ago #OnThisDay?

cc @DynamicsSIAM @emulenews @evamirandag
@johnmdudley @stevenstrogatz
#solitons #PDEs #integrablesystems https://t.co/Iruo5Vbgb6

[Academe, Math & More @academath]

ICYMI: New @Springermath #dynsys book

https://t.co/o9Xtl2D90S

[SpringerMath @SpringerMath]

📘 New from SpringerMath!
Averaging for Nonlinear Dynamics by Ferdinand Verhulst—linking averaging theory with real applications in physics & engineering, plus numerical bifurcations.
🔗 https://t.co/7xSQAkFOOK
#NonlinearDynamics #MathBooks #SpringerMath

[Integrable Systems, Nonlinear Waves & All That @integrablesys]

ICYMI: Updated book draft

#dynsys #mechanics #ODEs

https://t.co/R5SXIlBKvK

[COMSOL @COMSOL_Inc]

While watching a horse-drawn boat in a canal, #JohnScottRussell noticed a wave behaving strangely and followed it on horseback.

See how this led to the KdV equation and #soliton research: https://t.co/YYtWFq0QpH

[Academe, Math & More @academath]

💡 Gustav de Vries, known for the Korteweg-de Vries (#KdV) equation, one of the most important #integrablesystems having famous #soliton solutions, was born 160 years ago #onthisday

cc @adhara_mathphys @Aobeka11 @DynamicsSIAM @polbalen @StephenABoydPh1 https://t.co/yPwoUJsqGd

[Gabriel Peyré @gabrielpeyre]

Nonlinearity matters. Linear diffusion (heat) has non-compactly supported solutions. Non-linear diffusion (porous medium) drives dynamics with compactly supported solutions. The porous medium is the simplest case, studied in detail by Otto. https://t.co/rx9jkigoZa

[Integrable Systems, Nonlinear Waves & All That @integrablesys]

ICYMI: Special Issue: A Panorama of Nonlinear Dynamics and Complex Systems
#dynsys #chaos #freeaccess

https://t.co/5ZlNd1SVmb

[Gabriel Peyré @gabrielpeyre]

Nonlinearity matters. Linear diffusion (heat) has non-compactly supported solutions. Non-linear diffusion (porous medium) drives dynamics with compactly supported solutions. The porous medium is the simplest case, studied in details by Otto. https://t.co/rx9jkifR9C

[Academe, Math & More @academath]

💡 Edward Routh, known inter alia for ignorable coordinates & the Routhian in mechanics, was born 195 years ago #onthiday

https://t.co/x9sBDe9yMI

cc @adhara_mathphys @diff_eq @DynamicsSIAM @emulenews @_kmt46 @marianojavierd1 @polbalen @Sgabh @VergaraLautaro @witten_stein @wkbme

[Alec Helbling @alec_helbling]

In vector calculus, divergence quantifies the rate at which a field flows outward from a point.

A "source" (positive divergence) has net outgoing flow, a "sink" (negative divergence) net inward flow. A point with zero divergence is "incompressible" and has net zero flow.

[Probability and Statistics @probnstat]

Gradient descent is an iterative method for finding the minimum of a function by repeatedly moving in the direction that most steeply decreases its value. Starting from an initial guess, it uses derivatives to decide how to update parameters step by step, gradually improving a solution. In probability, gradient descent is used to maximize likelihoods and minimize divergences, allowing us to estimate parameters of statistical models from data. In machine learning, it is the engine behind training neural networks, logistic regression, and many other models, adjusting millions of weights so that predictions better match observed outcomes. In real life, the same idea appears whenever systems improve through feedback: businesses tuning prices, engineers optimizing designs, or even individuals refining habits. All follow a gradient-like process of making small, informed adjustments that steadily move toward better performance.

Image: https://t.co/ujvmvCJnQk

[Integrable Systems, Nonlinear Waves & All That @integrablesys]

📘 New @SpringerMath book on #integrablesystems https://t.co/pjcbhWkdlG

[SIAM Activity Group on Dynamical Systems @DynamicsSIAM]

Pedagogical article: "Discrete versus continuous -- lattice models and their exact continuous counterparts" (by Lorenzo Fusi, Oliver Křenek, Vít Průša, Casey Rodriguez, Rebecca Tozzi, Martin Vejvoda): https://t.co/SKVxsVO2Ds

[Frank Nielsen @FrnkNlsn]

Book #OTD

"Symmetry Methods for Differential Equations: A Beginner's Guide"

see https://t.co/6ZJnHx49mY

publisher home page:
https://t.co/0JG6TvcxWq

[High Energy Physics - Theory Papers @TheoryPapers]

Integrable Systems

Maciej Dunajski
https://t.co/iYR9C1mUEc [𝚑𝚎𝚙-𝚝𝚑]

[SIAM Activity Group on Dynamical Systems @DynamicsSIAM]

Lecture notes: "Classical Algebraic Geometry and Discrete Integrable Systems" (by Gessica Alecci, Michele Graffeo, Alexander Stokes): https://t.co/mAlbs6h6HW

[Mathelirium @mathelirium]

We just watched Professor Arthur Mattuck kick off MIT’s ODE course with the one interpretation that most differential equations classes somehow postpone: An ordinary differential equation isn’t primarily a method hunt. It’s a geometric rule. You write

dy/dx = f(x,y),

and that right-hand side is literally telling you the slope your solution curve must have at each point (x,y).

So I built this animation as a visual companion to that first lecture. It draws the direction field (little line elements whose slope is f(x,y)) and then shows integral curves sliding through it...curves that are tangent to the field everywhere they go.

Two quick examples from the animation:

For dy/dx = −x/y, the slope field steers you onto circles x² + y² = R². You also see a subtle point that gets missed when everything is taught as y(x)...even when the curve exists smoothly, the graph y(x) may only exist on a limited interval (|x|<R for the upper semicircle).

For dy/dx = 1 + x − y, the isoclines (curves where the slope is constant) make the global behavior obvious...trajectories get funneled into a corridor and become asymptotic to the special solution y=x. You learn qualitative behavior without solving it the traditional way.

#DifferentialEquations #ODEs #MITOCW #VectorFields #MathAnimation #Mathematics

[Differential Eqns @diff_eq]

There are important non-linear PDEs with closed-form solutions https://t.co/ejGjRodMyj