integrable-systems.bigb

= Integrable systems
{wiki=Integrable_system}

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

= Sinh-Gordon equation
{title2=$u_{z\bar z}=\frac12\sinh(2u)$}
{parent=Integrable systems}
{wiki}

The sinh-Gordon equation is an integrable nonlinear wave equation. Under the one-dimensional reduction $u=u(x)$ with $z=x+iy$, it becomes
$$
u_{xx}=2\sinh(2u).
$$
Writing $\phi=2u$ gives $\phi''=4\sinh\phi$ and the first integral
$$
\frac12(\phi')^2-4\cosh\phi=\text{constant}.
$$

= Sine-Gordon equation
{title2=$u_{XT}=\sin u$}
{parent=Integrable systems}
{c}
{wiki}

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 systems>[integrable system].

= One-soliton solution of the sine-Gordon equation in light-cone coordinates
{parent=Sine-Gordon equation}
{c}

For every $l>0$,
$$
u(X,T)=4\arctan\exp\left(-2lX-\frac{T}{2l}\right)
$$
solves $u_{XT}=\sin u$. The scaling $(X,T)\mapsto(e^sX,e^{-s}T)$ transforms $l$ by multiplication with $e^{-s}$.

= Sine-Gordon breather
{parent=Sine-Gordon equation}
{c}
{wiki=Breather}

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.

= Focusing nonlinear Schrodinger equation
{parent=Integrable systems}
{c}
{wiki=Nonlinear_Schr%C3%B6dinger_equation}

In the normalization
$$
i\psi_t+\psi_{xx}+2|\psi|^2\psi=0,
$$
the positive cubic term focuses the field and permits bright solitons on zero background.

= Bright standing soliton of the focusing nonlinear Schrodinger equation
{parent=Focusing nonlinear Schrodinger equation}
{c}

For every $\kappa>0$ and $x_0\in\mathbb R$,
$$
\psi(x,t)=\kappa e^{i\kappa^2t}
\operatorname{sech}(\kappa(x-x_0))
$$
solves $i\psi_t+\psi_{xx}+2|\psi|^2\psi=0$ and decays rapidly as $|x|\to\infty$.

= Hamiltonian form of the cubic nonlinear Schrodinger equations
{parent=Integrable systems}
{c}

The focusing and defocusing equations
$$
i\psi_t+\psi_{xx}\mathbin{\pm}2|\psi|^2\psi=0
$$
have Hamiltonians
$$
H_\pm=\int_{\mathbb R}\left(|\psi_x|^2\mp|\psi|^4\right)dx
$$
under the convention $i\psi_t=\delta H_\pm/\delta\overline\psi$.

= Defocusing nonlinear Schrödinger equation
{parent=Integrable systems}
{c}
{wiki=Nonlinear_Schr%C3%B6dinger_equation}

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.

= No rapidly decaying standing wave for the defocusing cubic nonlinear Schrodinger equation
{parent=Defocusing nonlinear Schrödinger equation}
{c}

For $\psi=e^{-iEt}f(x)$ with real $f$, the defocusing equation $i\psi_t+\psi_{xx}-2|\psi|^2\psi=0$ has first integral
$$
(f')^2=f^4-Ef^2.
$$
A 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 $f$.

= Dark soliton
{parent=Defocusing nonlinear Schrödinger equation}
{wiki}

For the normalization
$$
-2i\psi_t=\psi_{xx}+(1-|\psi|^2)\psi,
$$
a dark soliton of speed $0\leq U<1/\sqrt2$ is
$$
\psi_0(\xi)=
\sqrt{1-2U^2}\,
\tanh\left(\frac{\sqrt{1-2U^2}}{\sqrt2}\xi\right)
+i\sqrt2U,
\qquad \xi=x-Ut.
$$

= Korteweg-De Vries equation
{parent=Integrable systems}
{c}
{wiki=Korteweg%E2%80%93De_Vries_equation}

The Korteweg-De Vries equation
$$
u_t-6uu_x+u_{xxx}=0
$$
is an integrable nonlinear dispersive equation. Its sign convention here admits negative solitary waves.

= KdV Schrodinger spectral problem
{title2=$-\phi_{xx}+u\phi=\lambda\phi$}
{parent=Korteweg-De Vries equation}
{c}

The inverse-scattering spectral problem associated with this KdV convention is
$$
-\phi_{xx}+u(x,t)\phi=\lambda(t)\phi.
$$
For
$$
Q=\phi_t+u_x\phi-2(u+2\lambda)\phi_x,
$$
differentiation gives the <Wronskian> identity
$$
\partial_x(\phi_xQ-\phi Q_x)
=\phi^2\bigl[\dot\lambda-(u_t+u_{xxx}-6uu_x)\bigr].
$$

= Isospectrality of the KdV discrete spectrum
{parent=KdV Schrodinger spectral problem}
{c}

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.

= Evolution of a KdV discrete norming constant
{title2=$c_n(t)=c_n(0)e^{4\kappa_n^3t}$}
{parent=Isospectrality of the KdV discrete spectrum}
{c}

If $\varphi_n(x,t)\sim c_n(t)e^{-\kappa_nx}$ as $x\to+\infty$, the time equation $Q=0$ gives
$$
c_n'(t)=4\kappa_n^3c_n(t),
$$
and hence $c_n(t)=c_n(0)e^{4\kappa_n^3t}$.

= Soliton
{parent=Korteweg-De Vries equation}
{wiki}

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.

= Inverse scattering transform
{parent=Integrable systems}
{wiki}

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

= Discrete scattering data
{parent=Inverse scattering transform}

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.

= Gelfand-Levitan-Marchenko equation
{parent=Inverse scattering transform}
{c}
{wiki=Marchenko_equation}

For one-dimensional inverse scattering, the Gelfand-Levitan-Marchenko equation
$$
K(x,y)+F(x+y)+\int_x^\infty K(x,z)F(z+y)\,dz=0
$$
recovers the transformation kernel $K$ from scattering data encoded by $F$. In the convention used for the Korteweg-De Vries equation above, the potential is
$$
u(x,t)=-2\frac{\partial}{\partial x}K(x,x;t).
$$

= Rank-one Gelfand-Levitan-Marchenko kernel
{parent=Gelfand-Levitan-Marchenko equation}

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.

= Poisson-commuting separated energies
{parent=Integrable systems}
{c}

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.

= Separable forced harmonic oscillators
{parent=Integrable systems}

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.

= Action variable of a shifted harmonic oscillator
{parent=Integrable systems}

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

= Phase-plane area formula for an action variable
{parent=Action variable of a shifted harmonic oscillator}

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

= Compatibility condition for an overdetermined linear system
{parent=Integrable systems}

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

= Lax pair
{parent=Integrable systems}
{c}
{wiki}

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

= Airy equation
{title2=$q_t+q_{xxx}=0$}
{parent=Lax pair}
{c}
{wiki=Airy_function#Evolution_equation}

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 $e^{ik^3t}$.

= AKNS Lax pair for the nonlinear Schrodinger equation
{parent=Lax pair}
{c}

For complex fields $q,r$, the standard two-by-two AKNS pair has zero-curvature equations
$$
ir_t+r_{xx}+2qr^2=0,
\qquad
iq_t-q_{xx}-2rq^2=0.
$$
The reductions $q=\overline r$ and $q=-\overline r$ give the focusing and defocusing cubic nonlinear Schrödinger equations, respectively.

= Isospectral Lax equation
{parent=Lax pair}

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

= Trace invariants of a Lax equation
{parent=Isospectral Lax equation}

If
$$
\dot L=[A,L],
$$
then every positive power has constant trace:
$$
\frac d{dt}\operatorname{tr}(L^n)
=n\operatorname{tr}(L^{n-1}[A,L])=0.
$$
The final equality is the cyclic property of the <matrix trace>.

= Periodic KdV transfer matrix
{parent=Isospectral Lax equation}

For real periodic KdV data, translation by one period acts on a conjugate basis of scattering solutions by
$$
T=\begin{pmatrix}a&b\\\bar b&\bar a\end{pmatrix},
\qquad |a|^2-|b|^2=1.
$$
Lax compatibility gives $\dot T=[\Lambda,T]$ for the matrix of $\partial_t+A$ on the eigenspace. Hence $\operatorname{tr}T=2\operatorname{Re}a$ is conserved.

= Zero-curvature condition
{parent=Lax pair}
{wiki=Zero-curvature_condition}

For $\Phi_x+U\Phi=0$ and $\Phi_y+V\Phi=0$, compatibility is $V_x-U_y+[U,V]=0$.

= Global relation for the half-line free Schrodinger equation
{parent=Lax pair}

For $u_t=iu_{xx}$ on $x>0$, the half-line Fourier transform satisfies
$$
\widehat u_t+ik^2\widehat u=kh(t)-iu_x(0,t).
$$
Subtracting 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 $G(k,t)=kh(t)$.

= Tzitzeica equation
{parent=Integrable systems}
{c}
{wiki=Tzitzeica_equation}

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