It is known that the Korteweg-de Vries equation $$u_{t}+uu_{x}+u_{xxx} = 0,$$ with $u=u(x,t)$ smooth and with period equal to $L$, has important conservation laws, namely, $$E(u)=\frac{1}{2}\int_{0}^{L} u_{x}^{2} - \frac{1}{3}u^{3}\ dx\ \ \text{and}\ \ F(u)=\frac{1}{2}\int_{0}^{L}u^{2}\ dx.$$ $E...

For infinite dimensional integrable systems, there are usually infinite symmetries and conservation laws. For example, the KdV equation, the KP equation. However, unlike the classical symmetries (point symmetries), the higher symmetries (or Lie-Backlund symmetries; such as KdV hierarchy) seem us...

What is a good reference for a geometric version of Noether's theorem about Lagrangians, symmetries and conserved currents?

In most of the resources that I have read, integrable systems described by a PDE posses a zero-curvature equation $$ \partial_t U - \partial_x V + [U,V] = 0 $$ which gives rise to the monodromy matrix $$ T = \mathcal{P} \exp \int\limits_0^{L} \mathrm{d} x \; U $$ Then the following quantities $$ ...

I am looking for an exact solution for the following special case of Chini Equation with $2\geq a > 1 > b > 0, x, y \in \mathbb{R}^+$, $$\frac{dy}{dx} = 1 + \frac{a}{y} + \frac{b}{x}$$ I have tried to approach this using multiple methods and substitutions, but none has gotten me far. I know tha...

Would anybody be able to share a Mathematica/Matlab/other script for calculating Onsager's exact solution for the magnetisation of the 2d Ising model? I would be most grateful of one in order to test my MC simulations of the system.

I am looking for an exact solution to equation: $w''=aw^{-1/3}+b[f(y)]w^{-5/3}$, where $w=w(y), f(y)$ - arbitrary function (in this case $y^n$ with arbitrary $n$); $a,b$ - constants. Of course I can solve it using numerical methods for certain initial conditions, but I was looking for a exact s...

The nonlinear pde $$ \partial_t^2\phi-\partial_x^2\phi+\lambda\phi^3=0 $$ has the exact solution $$ \phi(x,t)=\mu\left(\frac{2}{\lambda}\right)^\frac{1}{4}{\rm sn}(p_0t-p\cdot x+\varphi,i) $$ with $\mu$ and $\varphi$ two integration constants and sn the snoidal Jacobi function, provided the...

It is common to solve PDEs with e.g. Fourier and Laplace Transforms. It is often said that Wavelets are a progression compared to them with many nice features. My question: Which Ansätze do you know to solve PDEs with Wavelets? Are these solution methods actually superior to the classical Ansätz...

Question I am having trouble trying to find a matrix $T$ so that with $X$, they form a Lax pair for the modified KdV equation $u_t - 6 u^2 u_x + u_{xxx} = 0$. Where $X$ is defined as: $ X = \begin{pmatrix} \lambda & i u\\ - i u & - \lambda\\ \end{pmatrix}$ I have been told that $T_{22} =...

I'm trying to understand the discussion around equation (2.1) of the paper http://www.jstor.org/stable/53053. It says that the linear PDE $M(\partial_x,\partial_y)q=0$ with constant coefficients has the Lax pair $\mu_x+ik\mu=q$ and $M(\partial_x,\partial_y)\mu=0$, where k is any complex number an...

I'm trying to understand what the conditions are for the Lax pairs for the zero-curvature representation: $$ \partial_t U - \partial_x V + [U,V]=0 $$ where $U=U(x,t,\lambda)$ and $V=V(x,t,\lambda)$ are matrix-valued functions and $\lambda$ is a parameter. The motivation behind this question is...

I have a second order differential equation given by: $x''(t) = \displaystyle\frac{\exp(-\frac{x(t)^2}{4t})}{A \sqrt{t}}$ I would like to be able to obtain an analytic solution to this equation, which may not be easy, but it's worth asking if anyone has any ideas! Frustratingly if instead of th...