Integrable systems

Integrable systems admit unusually rich exact structure such as Lax pairs, conserved quantities, and symmetry reductions.

The sinh-Gordon equation is an integrable nonlinear wave equation. Under the one-dimensional reduction $u=u(x)$ with $z=x+iy$, it becomesWriting $\varphi =2u$ gives ${\varphi }^{\prime \prime }=4\sinh \varphi$ and the first integral

In light-cone coordinates, the sine-Gordon equation can be written $u_{XT}=\sin u$. Its inverse scattering transform, solitons, breathers, and scaling symmetries make it an integrable system.

For every $l>0$,solves $u_{XT}=\sin u$. The scaling $(X,T)\mapsto (e^{s}X,e^{-s}T)$ transforms $l$ by multiplication with $e^{-s}$.

A sine-Gordon breather is a spatially localized solution that is periodic in time and can be interpreted through a complex-conjugate pair of discrete scattering eigenvalues.

In the normalizationthe positive cubic term focuses the field and permits bright solitons on zero background.

For every $\kappa >0$ and $x_0\in \mathbb{R}$,solves $i{\psi }_t+{\psi }_{xx}+2|\psi {|}^2\psi =0$ and decays rapidly as $|x|\to \infty$.

The focusing and defocusing equationshave Hamiltoniansunder the convention $i{\psi }_t=\delta H_{\pm }/\delta \overline{\psi }$.

Writing a defocusing nonlinear Schrödinger field as $\psi =\sqrt{\rho }{e}^{iS}$ turns it into a continuity equation for $\rho$ and a velocity equation for $v=S_x$, including a quantum-pressure term.

For $\psi =e^{-iEt}f(x)$ with real $f$, the defocusing equation $i{\psi }_t+{\psi }_{xx}-2|\psi {|}^2\psi =0$ has first integralA nonzero smooth function tending to zero at both infinities cannot satisfy this identity: a nonzero extremum requires $f^2=E>0$, but then the right side is negative for sufficiently small nonzero $f$.

For the normalizationa dark soliton of speed $0\le U<1/\sqrt{2}$ is

The Korteweg-De Vries equationis an integrable nonlinear dispersive equation. Its sign convention here admits negative solitary waves.

The inverse-scattering spectral problem associated with this KdV convention isFordifferentiation gives the Wronskian identity

For a rapidly decreasing KdV potential and a normalized bound-state eigenfunction, integration of the Wronskian identity over the real line gives ${\dot{\lambda }}_n=0$. Thus every discrete eigenvalue ${\lambda }_n=-{\kappa }_n^2$ is constant.

If ${\phi }_n(x,t)\sim c_n(t)e^{-{\kappa }_{n}x}$ as $x\to +\infty$, the time equation $Q=0$ givesand hence $c_n(t)=c_n(0)e^{4{\kappa }_n^{3}t}$.

A soliton is a localized travelling wave whose shape is preserved by the evolution and whose interactions with other solitons are elastic up to phase shifts.

The inverse scattering transform evolves the scattering data of an auxiliary linear operator and reconstructs the potential from the evolved data.

Discrete scattering data consist of the discrete eigenvalues and norming constants associated with bound states of the auxiliary spectral problem. Their simple time evolution reconstructs solitons and breathers through the inverse problem.

For one-dimensional inverse scattering, the Gelfand-Levitan-Marchenko equationrecovers the transformation kernel $K$ from scattering data encoded by $F$. In the convention used for the Korteweg-De Vries equation above, the potential is

If $F(s)=c{e}^{-\chi s}$ with $\chi >0$, the Gelfand-Levitan-Marchenko equation has a separable kernel. Writing $K(x,y)=-B(x)e^{-\chi y}$ reduces the integral equation to one algebraic equation for $B(x)$ and reconstructs a single soliton.

Hamiltonians that split into terms $F_i(p_i,q_i)$ have $\{F_i,F_j\}=0$ for $i\ne j$, and their differentials are generically independent because they use disjoint coordinate pairs.

A Hamiltonian that is a sum of one-coordinate quadratic potentials plus linear terms becomes a collection of independent harmonic oscillators after translating each equilibrium position.

For $F=(p^2+W^2{q}^2+aq)/2$, shifting $Q=q+a/(2W^2)$ gives oscillator energy $E=F+a^2/(8W^2)$ and action $I=E/|W|$.

The action $(2\pi {)}^{-1}\oint pdq$ is the area enclosed by a periodic phase-plane orbit divided by $2\pi$.

Commuting mixed derivatives impose an equation on the coefficient matrices of an overdetermined auxiliary system.

A Lax pair encodes a nonlinear equation as compatibility of two linear equations depending on an auxiliary spectral parameter.

The Airy equation is the linear dispersive partial differential equation $q_t+q_{xxx}=0$. Under the spatial Fourier transform, each mode evolves by multiplication by $e^{i{k}^{3}t}$.

For complex fields $q,r$, the standard two-by-two AKNS pair has zero-curvature equationsThe reductions $q=\overline{r}$ and $q=-\overline{r}$ give the focusing and defocusing cubic nonlinear Schrödinger equations, respectively.

For $\dot{L}=[L,A]$, the operator ${\partial }_t+A$ maps each eigenspace of $L$ into itself, so the spectrum is time-independent.

Ifthen every positive power has constant trace:The final equality is the cyclic property of the matrix trace.

For real periodic KdV data, translation by one period acts on a conjugate basis of scattering solutions byLax compatibility gives $\dot{T}=[\Lambda ,T]$ for the matrix of ${\partial }_t+A$ on the eigenspace. Hence $\mathrm{tr}T=2\mathrm{Re}a$ is conserved.

For ${\Phi }_x+U\Phi =0$ and ${\Phi }_y+V\Phi =0$, compatibility is $V_x-U_y+[U,V]=0$.

For $u_t=i{u}_{xx}$ on $x>0$, the half-line Fourier transform satisfiesSubtracting the relation at $-k$ cancels the unknown Neumann datum. Fourier inversion of the resulting odd combination gives the solution in terms of the initial transform and the prescribed Dirichlet datum, with boundary kernel $G(k,t)=kh(t)$.

The Tzitzeica equation in light-cone coordinates is $u_{xy}=e^u-e^{-2u}$ and admits a matrix Lax pair.