Integrable systems

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

A parameter-dependent system of differential relations maps a solution of one differential equation to a solution of another, or to a new solution of the same equation. Compatibility of the relations supplies the equations of motion. Integration constants select a particular transformed field. The Sine-Gordon Bäcklund transformation is an example of an auto-transformation and also generates conserved currents.

For a finite-dimensional Hamiltonian system, integrability is normally expressed by enough independent first integrals in mutual Poisson bracket involution. In classical field theories, an infinite hierarchy of compatible conserved quantities is the corresponding structure. For Sine-Gordon theory, Bäcklund transformations generate local conservation laws and explicit soliton solutions. Conservation of arbitrary quantities alone should not be confused with proof of their Hamiltonian involution.

An exact-solution method that transforms a nonlinear partial differential equation into equations bilinear in auxiliary Hirota tau functions. Finite exponential expansions of the Hirota tau functions produce soliton families. The auxiliary exponential coefficients need not all be positive.

An auxiliary function used in Hirota's bilinear method to encode a solution of an integrable partial differential equation. Products and ratios of tau functions replace nonlinear field variables; their differential equations become bilinear. In a Sine-Gordon multisoliton tau representation, two real tau functions encode the field as a continuous $4\arg (f+ig)$.

The periodic Hamiltonian chain $H=\frac{1}{2}\sum p_i^2+\sum e^{q_i-q_{i+1}}$, with cyclic indices, is a Liouville integrable system. Exponentials of neighbouring coordinate differences lead to a Lax pair and conserved characteristic coefficients.

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.

The transformationsform a one-parameter Lie point symmetry of $u_{XT}=\sin u$. The product $z=XT$ is a similarity variable, and a group-invariant solution $u=F(z)$ satisfies

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.

The third Painlevé equation is the nonlinear second-order ordinary differential equationThe substitution $w=e^{iF}$ in the scaling reduction of the sine-Gordon equation gives the parameter choice $(\alpha ,\beta ,\gamma ,\delta )=(1/2,-1/2,0,0)$.

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

A Jost solution of a one-dimensional Schrodinger equation is specified by plane-wave asymptotics at one spatial infinity. Comparing the left and right Jost solutions defines transmission and reflection coefficients.

For a rapidly decaying KdV potential, the scattering data consist of the continuous reflection coefficient, the negative discrete eigenvalues $-{\kappa }_n^2$, and a positive norming constant for each bound state.

For $u_t-6u{u}_x+u_{xxx}=0$, the discrete eigenvalues are fixed, whilewhen $c_n$ denotes the bound-state asymptotic amplitude. The squared amplitudes appearing in the Marchenko kernel consequently evolve by $e^{8{\kappa }_n^{3}t}$.

For a weak potential $u=ϵq$, the first Born approximation in the KdV Schrödinger convention is

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.

An exact nonlinear-wave solution containing several localized solitons. In an integrable scattering solution, these objects approach separated incoming and outgoing profiles, with collision effects encoded by position shifts and internal phases. Bäcklund transformations and Hirota's bilinear method construct examples such as the Sine-Gordon multisoliton tau representation.

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.

An integrable partial differential equation has enough exact structure, such as a Lax pair, infinitely many conserved quantities, or an inverse scattering transform, to reduce its evolution to auxiliary linear or finite-dimensional data.

The Harry Dym equation is the integrable partial differential equation $p_t=p^3{p}_{xxx}$. It admits a Lax pair based on the second-order operator $L=-{\partial }_x{p}^2{\partial }_x$.

The Boussinesq equation is a nonlinear dispersive wave equation. One integrable normalization is

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

For symmetric $L$ and antisymmetric $A$ satisfying $\dot{L}=LA-AL$, solve $\dot{Q}=-AQ$, $Q(0)=I$. Then $Q$ is orthogonal and $Q^{T}LQ$ is constant. Hence every eigenvalue and every coefficient of the characteristic polynomial of $L$ is conserved.

For periodic $U(x,t,\lambda )$ with period $L$, solve $W_x=UW$, $W(0)=I$. The monodromy matrix is $w=W(L)$. If $U,V$ satisfy the zero-curvature condition and are periodic, then $W_t=VW-WV(0)$, hence $w_t=[V(0),w]$. The matrix trace of each power of $w$ is a first integral.

A spectral parameter is an auxiliary complex parameter in a linear system whose compatibility encodes a nonlinear integrable equation. Its analytic dependence organizes conserved quantities and scattering data.

For covariant derivatives on Euclidean four-space, one anti-self-dual Yang-Mills Lax pair isThe condition $[L,M]=0$ for every $\zeta \in {\mathbb{CP}}^1$ is equivalent to anti-self-duality of the curvature.

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 $e^{i{k}^{3}t}$.

The scalar linear partial differential equation $u_t+u_x+u_{xxx}=0$ is an Airy equation with a first-order transport term. It is a linearization of the Korteweg-De Vries equation with an advective term retained. It is distinct from the viscous incompressible-flow Stokes equation. On the whole line its spatial Fourier transform evolves by $e^{i(k^3-k)t}$.

For a time Laplace transform parameter with $\mathrm{Re}p>0$, the spatial characteristic roots satisfy $r^3+r+p=0$. Exactly one root $r_{-}$ has negative real part and two have positive real part. Indeed an imaginary root would force $p=-i(k-k^3)$ to be imaginary. The count is therefore constant on the connected right half-plane; for real $p>0$, the cubic has one negative real root and a conjugate pair with positive real parts. Repeated roots would satisfy $3r^2+1=0$, which again puts $p$ on the imaginary axis. Thus $r_{-}(p)$ is a nonzero holomorphic function in the right half-plane. Only $e^{r_{-}x}$ decays at positive spatial infinity.

For $\mathrm{Re}p>0$, let $r_{-}$ and $r_1,r_2$ be the characteristic roots of $r^3+r+p=0$, split by the signs of their real parts. Put $d_j=3r_j^2+1$. The real-line Green function of $p+{\partial }_x+{\partial }_x^3$ isThe reciprocal-polynomial root identities imply continuity of $R_p,R_p^{\prime }$ and jump $R_p^{\prime \prime }(0+)-R_p^{\prime \prime }(0-)=1$. The distributional derivative therefore gives $(p+{\partial }_s+{\partial }_s^3)R_p={\delta }_0$. The transport term changes the roots: the pure Airy resolvent kernel with $r^3+p=0$ cannot be substituted unchanged.

For the linear dispersive Stokes equation on $x>0$, with sufficiently smooth decaying initial data and $\mathrm{Re}p>0$, define $V(x,p)={\int }_0^{\infty }{R}_p(x-y)u_0(y)dy$ and let $G(p)$ be the time Laplace transform of the prescribed derivative $u_x(0,t)$. Spatial decay and the one-dimensional decaying homogeneous space giveThe Neumann boundary condition follows by differentiating at zero. The initial forcing follows from the Stokes resolvent kernel with advection. The Bromwich inversion formula gives an integral representation with no unknown boundary traces. Truncating or extending the prescribed boundary signal beyond a target time does not affect earlier values, by the Laplace transform time-shift rule and causality of this Green function.

For $\mathrm{Re}p>0$, put $\lambda =p^{1/3}$ using the principal cube root and $r_{1,2}=\lambda e^{\pm i\pi /3}$. The decaying Green function for ${\partial }_s^3+p$ on the real line isThe identities among the three characteristic roots show that $R_p$ and $R_p^{\prime }$ are continuous, while $R_p^{\prime \prime }$ has jump one. Its distributional derivative therefore satisfies $({\partial }_s^3+p)R_p={\delta }_0$. This is the kernel of the resolvent operator $(p+{\partial }_s^3{)}^{-1}$, obtained by the time Laplace transform of the Airy equation.

For $u_t+u_{xxx}=0$ on $x>0$, define $V(x,p)={\int }_0^{\infty }{R}_p(x-y)u_0(y)dy$ and let $G$ be the time Laplace transform of $u_x(0,t)$. The transformed solution with spatial decay isOnly one characteristic root has negative real part, so one scalar boundary derivative fixes the remaining homogeneous mode. The Bromwich inversion formula supplies an integral representation involving only the initial and boundary data. For general linear differential equations on a half-line, the number and form of the necessary boundary data depend on the decaying spatial roots; Fokas and Wang study the corresponding boundary maps for linear dispersive equations.

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.

The Lax equation $L_t=[L,A]$ for these operators is equivalent toEliminating $p$ gives the Boussinesq equation

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

With boundary values $q_0=q_n=0$, the finite Volterra lattice isup to rescaling or reversal of time. Writing $q_j=a_j^2$ gives ${\dot{a}}_j=a_j(a_{j+1}^2-a_{j-1}^2)$.

Let $L$ be the symmetric zero-diagonal Jacobi matrix with $L_{j,j+1}=L_{j+1,j}=a_j$, and let the skew-symmetric matrix $B$ haveThen $\dot{L}=[B,L]$ is equivalent to the Finite Volterra lattice. The corresponding similarity flow makes the eigenvalues of $L$, the coefficients of its characteristic polynomial, and its trace invariants first integrals.

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.