09.06.04 · classical-mech / integrable

Lax Pairs and the Inverse Scattering Method: Solitons in Classical Mechanics

shipped3 tiersLean: nonepending prereqs

Anchor (Master): Faddeev & Takhtajan, Hamiltonian Methods in the Theory of Solitons (2007); Novikov et al., Theory of Solitons (1984); Ablowitz & Segur, Solitons and the Inverse Scattering Transform (1981)

Intuition Beginner

A soliton is a wave that does not break up. In most wave systems, a localised pulse disperses: different frequency components travel at different speeds, and the pulse flattens out and disappears. A soliton is different. Nonlinear effects exactly cancel dispersion, and the pulse maintains its shape indefinitely. When two solitons collide, they pass through each other and re-emerge with their original shapes and speeds, as if nothing had happened. The only trace of the collision is a small shift in position -- a "phase shift" -- that records the encounter ^{[source pending]}.

The discovery was accidental. In 1834, the Scottish engineer John Scott Russell was riding alongside the Union Canal near Edinburgh when a barge stopped suddenly. He watched a rounded heap of water detach from the bow, travel down the canal without changing shape, and continue for miles. He chased it on horseback. This single wave of elevation -- the first recorded soliton -- became the seed of a major branch of mathematical physics ^{[source pending]}.

The mathematical explanation waited until 1965. Norman Zabusky and Martin Kruskal were running computer simulations of the Korteweg-de Vries (KdV) equation, a partial differential equation that models shallow water waves, written down by Diederik Korteweg and Gustav de Vries in 1895. Zabusky and Kruskal started with an initial pulse that evolved into several localised humps of different heights. They watched, astonished, as the humps passed through each other and emerged with their original shapes intact. Each hump was a solitary wave -- a soliton. The word was coined by Zabusky and Kruskal to emphasise the particle-like nature of these waves: they collide and scatter like particles, preserving their identity ^{[source pending]}.

What makes this surprising is that the KdV equation is nonlinear. Linear waves superpose: you just add them up, and the result is the sum of the parts. Nonlinear waves interact in complicated ways that usually produce new frequencies, dispersion, and eventually chaos. The fact that KdV solitons survive their collisions intact was deeply unexpected. The explanation came in two stages. First, in 1967, Clifford Gardner, John Greene, Martin Kruskal, and Robert Miura discovered that the KdV equation can be solved by a three-step procedure they called the inverse scattering transform (IST) ^{[source pending]}. Then, in 1968, Peter Lax reformulated the method in an elegant algebraic framework: the Lax pair ^{[source pending]}.

The key idea is to encode the nonlinear dynamics as a linear problem. Instead of solving the KdV equation directly, you represent the solution $u (x, t)$ as a potential in a linear operator $L$ -- the same Schrodinger operator that appears in quantum mechanics. You find a second operator $M$ such that the equation $d L / d t = [M, L] = M L - L M$ is exactly equivalent to the KdV equation. The eigenvalues of $L$ are then constants of the motion: they do not change as the potential evolves. This property is called isospectral evolution -- the "spectrum" (set of eigenvalues) stays fixed.

The IST works in three steps, analogous to how the Fourier transform solves the heat equation:

Scatter. At the initial time $t = 0$ , treat $u (x, 0)$ as a potential and solve the quantum scattering problem. This produces the "scattering data": bound-state eigenvalues and normalisation constants, plus a reflection coefficient.
Evolve. The scattering data evolves in a simple, linear way. The bound-state eigenvalues are constant (isospectral), and the other quantities evolve by elementary exponential time dependence.
Reconstruct. Solve an inverse problem: given the time-evolved scattering data, reconstruct the potential $u (x, t)$ . This inverse problem is a linear integral equation.

The entire nonlinear dynamics of the KdV equation has been reduced to three linear steps. The soliton solutions correspond to the simplest scattering data: a finite number of bound states with no reflected wave. Each bound state produces one soliton, and an initial condition with $N$ bound states evolves into $N$ solitons that collide and separate.

Visual Beginner

Figure 1. Two KdV solitons colliding. The taller (faster) soliton overtakes the shorter one. During the collision the wave profile is complicated, but after the collision both solitons re-emerge unchanged except for a position shift.

Soliton property	Description
Shape preservation	Each pulse maintains its width and amplitude
Speed--amplitude relation	Taller solitons travel faster
Elastic collision	After passing through each other, shapes are unchanged
Phase shift	Only effect of collision is a position offset

The balance between nonlinearity and dispersion is delicate. If the nonlinear term is too weak, the pulse disperses. If it is too strong, the pulse steepens and breaks. The soliton exists at the exact tipping point where these two effects cancel.

Worked example Beginner

The one-soliton solution of KdV.

The KdV equation is $u_{t} + 6 u u_{x} + u_{xxx} = 0$ , where subscripts denote derivatives. The one-soliton solution is

u (x, t) = 2 κ^{2} sech^{2} (κ (x - 4 κ^{2} t - x_{0})),

where $κ > 0$ is a parameter and $x_{0}$ is the initial position ^{[source pending]}.

The function $sech (z) = 2/ (e^{z} + e^{- z})$ is a bell-shaped curve centred at $z = 0$ with maximum value $1$ . The parameter $κ$ controls both the amplitude ( $2 κ^{2}$ ) and the speed ( $4 κ^{2}$ ): taller solitons travel faster. The width is proportional to $1/ κ$ : taller solitons are narrower.

At $t = 0$ , the soliton is centred at $x = x_{0}$ . At time $t$ , it has moved to $x = x_{0} + 4 κ^{2} t$ . The profile does not change shape. This is a travelling wave: a wave that moves without dispersing.

For $κ = 1$ and $x_{0} = 0$ : $u (x, t) = 2 sech^{2} (x - 4 t)$ . The peak height is $2$ , the speed is $4$ , and the width (distance over which the profile drops to half its peak) is about $1.5$ .

For $κ = 0.5$ : $u (x, t) = 0.5 sech^{2} ((x - t) /2)$ . The peak is $0.5$ and the speed is $1$ -- slower and lower. If both solitons start near $x = 0$ at $t = 0$ , the faster one ( $κ = 1$ ) gradually pulls ahead. At large times the two are well separated and each has its original shape.

Check your understanding Beginner

Formal definition: the Lax pair and inverse scattering Intermediate+

The KdV equation

The Korteweg-de Vries equation in standard nondimensional form is

u_{t} + 6 u u_{x} + u_{xxx} = 0.

It models long waves in shallow water, where $u (x, t)$ is the deviation of the surface from equilibrium. The term $u_{xxx}$ represents dispersion: different wavelengths travel at different speeds, which tends to spread out a wave packet. The term $6 u u_{x}$ represents nonlinear steepening: taller parts of the wave move faster, which tends to sharpen the wave front. The balance between these two opposing effects is what produces solitons ^{[source pending]}.

The KdV equation is a Hamiltonian system on an infinite-dimensional phase space. The phase space is a space of functions $u (x)$ (decaying sufficiently fast as $x \to \pm \infty$ ), equipped with the Poisson bracket

{F, G} = \int_{- \infty}^{\infty} \frac{δ F}{δ u} \frac{d}{d x} \frac{δ G}{δ u} d x,

where $δ F / δ u$ is the functional derivative. The Hamiltonian is

H [u] = \int_{- \infty}^{\infty} (\frac{1}{2} u_{x}^{2} - u^{3}) d x,

and the KdV equation is equivalent to $u_{t} = {H, u}$ under this bracket.

Key result: Lax pairs — the central definition

Definition (Lax pair). Let $L (t)$ and $M (t)$ be linear operators on a Hilbert space (or a finite-dimensional vector space) depending on time $t$ . The pair $(L, M)$ is a Lax pair for an evolution equation if the equation is equivalent to

\frac{d L}{d t} = [M, L] = M L - L M .

The operator $L$ is the Lax operator and $M$ is the partner (or isomonodromic operator). The equation $d L / d t = [M, L]$ is the Lax equation ^{[source pending]}.

The fundamental consequence is isospectral evolution:

Theorem (Isospectral evolution). If $L (t)$ satisfies $d L / d t = [M, L]$ , then the eigenvalues of $L (t)$ are independent of $t$ .

Proof. Let $U (t)$ be the solution of $d U / d t = M U$ with $U (0) = I$ . Then

L (t) = U (t) L (0) U (t)^{- 1} .

To verify, differentiate using the product rule and the identity $d (U^{- 1}) / d t = - U^{- 1} (d U / d t) U^{- 1} = - U^{- 1} M U^{- 1}$ :

\frac{d}{d t} (U^{- 1} LU) = (- U^{- 1} M U^{- 1}) LU + U^{- 1} \frac{d L}{d t} U + U^{- 1} L M U^{- 1} U = U^{- 1} (\frac{d L}{d t} - [M, L]) U = 0.

Since $L (t)$ is related to $L (0)$ by a similarity transformation, they have the same eigenvalues. $□$

The conserved quantities are the spectral invariants of $L$ : the traces $tr (L^{k})$ , the characteristic polynomial coefficients, and so on. For an infinite-dimensional operator, the spectrum itself is the conserved quantity. Since there are infinitely many spectral invariants, a Lax pair generates infinitely many conserved quantities -- the hallmark of complete integrability.

The Lax pair for KdV

Lax (1968) showed that the KdV equation has a Lax pair where $L$ is the one-dimensional Schrodinger operator:

L = - \frac{\partial ^{2}}{\partial x ^{2}} + u (x, t),

and $M$ is the third-order differential operator

M = - 4 \frac{\partial ^{3}}{\partial x ^{3}} + 3 (u \frac{\partial}{\partial x} + \frac{\partial}{\partial x} u) .

The Schrodinger operator $L$ is familiar from quantum mechanics: its eigenvalue problem $L y = λ y$ is

- y^{''} + u (x, t) y = λ y,

where the "potential" is the KdV solution $u (x, t)$ . The partner $M$ is constructed so that the Lax equation reproduces the KdV equation ^{[source pending]}.

To verify, compute $d L / d t = \partial u / \partial t$ (since $L$ depends on $t$ only through $u$ ). The commutator $[M, L]$ is a differential operator whose leading terms cancel (the highest-order derivatives in $M$ and $L$ commute), leaving a zeroth-order (multiplication) operator. Explicit computation gives

[M, L] = - u_{xxx} - 6 u u_{x} .

Therefore $d L / d t = [M, L]$ becomes $\partial u / \partial t = - u_{xxx} - 6 u u_{x}$ , which is $(- 1)$ times the KdV equation. The standard sign convention absorbs this factor.

The inverse scattering transform

The Lax pair reduces the KdV equation to the isospectral deformation of the Schrodinger operator. The inverse scattering transform (IST) exploits this to solve the initial-value problem ^{[source pending]}.

Step 1: Direct scattering. Given the initial data $u (x, 0)$ , solve the eigenvalue problem for the Schrodinger operator $L = - \partial^{2} / \partial x^{2} + u (x, 0)$ :

- y^{''} + u (x, 0) y = λ y, x \in (- \infty, \infty) .

For a potential $u (x, 0)$ that decays as $x \to \pm \infty$ , the spectrum has two parts:

Discrete spectrum (bound states): a finite number of negative eigenvalues $λ_{n} = - κ_{n}^{2} < 0$ , $n = 1, \dots, N$ , with localised eigenfunctions. Each bound state has a normalisation constant $c_{n}$ defined by the asymptotic behaviour $y_{n} (x) \sim c_{n} e^{- κ_{n} x}$ as $x \to + \infty$ .
Continuous spectrum: the positive real axis $λ = k^{2} > 0$ . For each $k$ , there is a solution with asymptotic behaviour $y \sim e^{- ik x} + r (k) e^{ik x}$ as $x \to + \infty$ and $y \sim t (k) e^{- ik x}$ as $x \to - \infty$ . The function $r (k)$ is the reflection coefficient and $t (k)$ is the transmission coefficient.

The scattering data is $S = {r (k), κ_{n}, c_{n} : n = 1, \dots, N}$ .

Step 2: Time evolution of scattering data. The Lax equation ensures the eigenvalues $κ_{n}$ are constant. The normalisation constants and reflection coefficient evolve as

c_{n} (t) = c_{n} (0) e^{4 κ_{n}^{3} t}, r (k, t) = r (k, 0) e^{8 i k^{3} t} .

These are elementary time dependences: the scattering data at any time $t$ is obtained from the initial scattering data by multiplying by exponentials.

Step 3: Inverse scattering (Gelfand-Levitan-Marchenko equation). From the time-evolved scattering data $S (t)$ , reconstruct the potential $u (x, t)$ by solving the linear integral equation

K (x, y) + F (x + y) + \int_{x}^{\infty} K (x, z) F (z + y) d z = 0, y > x,

where

F (ξ) = \frac{1}{2 π} \int_{- \infty}^{\infty} r (k, t) e^{ik ξ} d k + n = 1 \sum N c_{n} (t)^{2} e^{- κ_{n} ξ} .

The solution of the KdV equation is

u (x, t) = - 2 \frac{d}{d x} K (x, x) .

This three-step procedure -- scatter, evolve, reconstruct -- is the inverse scattering transform. It is the nonlinear analogue of the Fourier transform: decompose the initial data into "normal modes" (scattering data), evolve the modes independently, and reassemble ^{[source pending]}.

N-soliton solutions

The pure $N$ -soliton solution corresponds to the case $r (k) = 0$ (no reflected wave). The scattering data is ${κ_{1}, \dots, κ_{N}, c_{1}, \dots, c_{N}}$ . In this case the Gelfand-Levitan-Marchenko equation reduces to a finite-dimensional linear algebra problem. Define the $N \times N$ matrix $A$ with entries

A_{mn} = δ_{mn} + \frac{c _{m}^{2} c _{n}^{2}}{κ _{m} + κ _{n}} e^{- (κ_{m} + κ_{n}) x} .

The $N$ -soliton solution is

u (x, t) = - 2 \frac{d ^{2}}{d x ^{2}} lo g det A .

For $N = 1$ , this gives the one-soliton $u = 2 κ_{1}^{2} sech^{2} (κ_{1} (x - 4 κ_{1}^{2} t - x_{0}))$ with $x_{0} = - \frac{1}{2 κ _{1}} lo g (c_{1}^{2} / (2 κ_{1}))$ .

For $N = 2$ , the determinant is

det A = 1 + \frac{c _{1}^{2}}{2 κ _{1}} e^{- 2 κ_{1} x} + \frac{c _{2}^{2}}{2 κ _{2}} e^{- 2 κ_{2} x} + \frac{( κ _{1} - κ _{2} ) ^{2}}{( κ _{1} + κ _{2} ) ^{2}} \frac{c _{1}^{2} c _{2}^{2}}{4 κ _{1} κ _{2}} e^{- 2 (κ_{1} + κ_{2}) x} .

As $t \to \pm \infty$ , this decomposes into two separated one-soliton solutions. The faster soliton is shifted forward and the slower one is shifted backward. The phase shift of the $j$ -th soliton due to the collision is

Δ x_{j} = \frac{1}{κ _{j}} k \neq = j \sum lo g \frac{κ _{j} - κ _{k}}{κ _{j} + κ _{k}} .

The sign of the shift depends on the ordering of the $κ$ values ^{[source pending]}.

The Lax pair and conservation laws

The isospectral property generates an infinite sequence of conservation laws. The spectral invariants $tr (L^{k})$ are conserved for each $k$ , and these correspond to integrals of local densities. The first three are:

$I_{1} = \int u d x$ (mass, or "total elevation").
$I_{2} = \int u^{2} d x$ (momentum).
$I_{3} = \int (u^{3} - \frac{1}{2} u_{x}^{2}) d x$ (energy, proportional to the Hamiltonian).

The existence of infinitely many conservation laws is another hallmark of complete integrability. For a generic PDE, there are only finitely many independent conservation laws. The KdV equation has infinitely many because its Lax operator has infinitely many spectral invariants ^{[source pending]}.

The KdV equation also has a bi-Hamiltonian structure: it can be written using either of two compatible Poisson brackets, and the Lenard recursion between them generates the entire infinite hierarchy of conserved quantities and commuting flows. The first Poisson operator is $B_{1} = \partial / \partial x$ and the second is $B_{2} = \partial^{3} / \partial x^{3} + 2 u \partial / \partial x + 2 \partial / \partial x \cdot u$ . The KdV equation is $u_{t} = B_{1} δ H_{2} / δ u = B_{2} δ H_{1} / δ u$ , and the recursion $B_{1} δ H_{n + 1} / δ u = B_{2} δ H_{n} / δ u$ produces the infinite hierarchy. This bi-Hamiltonian structure is the algebraic mechanism that underlies integrability.

Exercises Intermediate+

Exercise 1 (medium, symbolic).

Verify that $u (x, t) = 2 sech^{2} (x - 4 t)$ satisfies the KdV equation $u_{t} + 6 u u_{x} + u_{xxx} = 0$ . Use $d (sech ξ) / d ξ = - sech ξ tanh ξ$ and $tanh^{2} ξ + sech^{2} ξ = 1$ .

Hint

Let $ξ = x - 4 t$ . Compute $u_{t}$ , $u_{x}$ , and $u_{xxx}$ in terms of $sech ξ$ and $tanh ξ$ , then substitute into the KdV equation.

Answer

$u = 2 sech^{2} ξ$ with $ξ = x - 4 t$ . Then:

$u_{t} = 2 \cdot 2 sech ξ (- sech ξ tanh ξ) (- 4) = 16 sech^{2} ξ tanh ξ$ .

$u_{x} = 2 \cdot 2 sech ξ (- sech ξ tanh ξ) (1) = - 4 sech^{2} ξ tanh ξ$ .

$6 u u_{x} = 6 \cdot 2 sech^{2} ξ \cdot (- 4 sech^{2} ξ tanh ξ) = - 48 sech^{4} ξ tanh ξ$ .

$u_{xx} = d / d ξ (- 4 sech^{2} ξ tanh ξ) = 8 sech^{2} ξ tanh^{2} ξ - 4 sech^{4} ξ$ . Using $tanh^{2} = 1 - sech^{2}$ : $u_{xx} = 8 sech^{2} ξ - 12 sech^{4} ξ$ .

$u_{xxx} = d / d ξ (8 sech^{2} ξ - 12 sech^{4} ξ) = - 16 sech^{2} ξ tanh ξ + 48 sech^{4} ξ tanh ξ$ .

Therefore $u_{t} + 6 u u_{x} + u_{xxx} = 16 sech^{2} ξ tanh ξ - 48 sech^{4} ξ tanh ξ - 16 sech^{2} ξ tanh ξ + 48 sech^{4} ξ tanh ξ = 0$ .

Exercise 3 (medium, symbolic).

For the one-soliton initial condition $u (x, 0) = 2 κ^{2} sech^{2} (κ x)$ , find the scattering data of the Schrodinger operator $L = - \partial^{2} / \partial x^{2} + u (x, 0)$ . How many bound states does it have, and what is the reflection coefficient?

Hint

The Schrodinger equation $- y^{''} + 2 κ^{2} sech^{2} (κ x) y = λ y$ can be solved exactly by the substitution $z = tanh (κ x)$ . The equation becomes an associated Legendre equation.

Answer

The potential $V (x) = - 2 κ^{2} sech^{2} (κ x)$ has the form $- ℓ (ℓ + 1) κ^{2} sech^{2} (κ x)$ with $ℓ (ℓ + 1) = 2$ , so $ℓ = 1$ . The standard result for this class of potentials is that there are exactly $ℓ$ bound states and the potential is reflectionless: $r (k) = 0$ .

For $ℓ = 1$ , there is exactly one bound state with eigenvalue $λ_{1} = - κ^{2}$ (i.e., $κ_{1} = κ$ ). The bound-state eigenfunction is $y_{1} \propto sech (κ x)$ , which has asymptotic behaviour $y_{1} \sim 2 e^{- κ x}$ as $x \to + \infty$ , giving normalisation constant $c_{1} = 2 κ^{1/2}$ (up to phase convention).

The scattering data is ${κ_{1} = κ, c_{1}, r (k) = 0}$ : one bound state, no reflection.

Exercise 4 (hard, symbolic).

Verify the KdV Lax pair by computing the commutator $[M, L]$ for $L = - \partial^{2} / \partial x^{2} + u$ and $M = - 4 \partial^{3} / \partial x^{3} + 3 (u \partial / \partial x + (\partial / \partial x) u)$ . Show that the result is the multiplication operator $- u_{xxx} - 6 u u_{x}$ .

Hint

Write $M = - 4 D^{3} + 6 u D + 3 u^{'}$ (using $D u = u D + u^{'}$ , so $3 (u D + D u) = 6 u D + 3 u^{'}$ ). Compute $[- 4 D^{3}, u]$ , $[6 u D, - D^{2}]$ , $[3 u^{'}, - D^{2}]$ , $[6 u D, u]$ , and $[3 u^{'}, u]$ separately, using the commutator identity $[D^{n}, u] = \sum_{j = 0}^{n - 1} (j n) u^{(n - j)} D^{j}$ .

Answer

Write $D = \partial / \partial x$ . Using $D u = u D + u^{'}$ , we have $3 (u D + D u) = 6 u D + 3 u^{'}$ , so $M = - 4 D^{3} + 6 u D + 3 u^{'}$ .

The relevant commutators with $L = - D^{2} + u$ are:

$[- 4 D^{3}, u] = - 4 (u^{'''} + 3 u^{''} D + 3 u^{'} D^{2})$ .

$[6 u D, - D^{2}]$ : Since $[u D, D^{2}] = - 2 u^{'} D^{2} - u^{''} D$ , we get $[6 u D, - D^{2}] = 12 u^{'} D^{2} + 6 u^{''} D$ .

$[3 u^{'}, - D^{2}]$ : Since $[u^{'}, D^{2}] = - u^{'''} - 2 u^{''} D$ (verified by acting on a test function), $[3 u^{'}, - D^{2}] = 3 u^{'''} + 6 u^{''} D$ .

$[6 u D, u] = 6 u u^{'}$ and $[3 u^{'}, u] = 0$ .

Collecting all terms:

$- 4 u^{'''} - 12 u^{''} D - 12 u^{'} D^{2} + 12 u^{'} D^{2} + 6 u^{''} D + 3 u^{'''} + 6 u^{''} D + 6 u u^{'}$ .

The $D^{2}$ terms: $- 12 u^{'} + 12 u^{'} = 0$ . The $D$ terms: $- 12 u^{''} + 6 u^{''} + 6 u^{''} = 0$ . Zeroth order: $- 4 u^{'''} + 3 u^{'''} + 6 u u^{'} = - u_{xxx} + 6 u u_{x}$ .

So $[M, L] = - (u_{xxx} + 6 u u_{x})$ , confirming $d L / d t = [M, L]$ reproduces KdV.

Exercise 5 (hard, symbolic).

From the two-soliton formula $det A = 1 + e^{η_{1}} + e^{η_{2}} + A_{12} e^{η_{1} + η_{2}}$ where $η_{j} = 2 κ_{j} x - 8 κ_{j}^{3} t + δ_{j}$ and $A_{12} = (κ_{1} - κ_{2})^{2} / (κ_{1} + κ_{2})^{2}$ , show that as $t \to + \infty$ the solution decomposes into two separated one-soliton solutions. Compute the phase shift of each soliton (assume $κ_{1} > κ_{2}$ ).

Hint

Near the faster soliton ( $x \approx 4 κ_{1}^{2} t$ ), $η_{1} = O (1)$ while $η_{2} \to - \infty$ . Factor out the dominant exponential. Near the slower soliton ( $x \approx 4 κ_{2}^{2} t$ ), $η_{2} = O (1)$ while $η_{1} \to + \infty$ .

Answer

Near the faster soliton ( $x \approx 4 κ_{1}^{2} t$ , so $η_{1} = O (1)$ ): Since $κ_{1} > κ_{2}$ ,

$η_{2} = 2 κ_{2} x - 8 κ_{2}^{3} t + δ_{2} \approx 8 κ_{2} (κ_{1}^{2} - κ_{2}^{2}) t + δ_{2} \cdot 2 κ_{2} \to - \infty$

as $t \to + \infty$ (because $η_{2}$ decreases at rate $8 κ_{2} (κ_{2}^{2} - κ_{1}^{2}) < 0$ ). So $e^{η_{2}} \to 0$ and the cross term $e^{η_{1} + η_{2}} \to 0$ . The determinant reduces to $det A \approx 1 + e^{η_{1}}$ , giving $u \approx 2 κ_{1}^{2} sech^{2} (η_{1} /2)$ .

Near the slower soliton ( $x \approx 4 κ_{2}^{2} t$ , so $η_{2} = O (1)$ ): Now $η_{1} \to + \infty$ , so $e^{η_{1}}$ dominates. Factor: $det A \approx e^{η_{1}} (1 + A_{12} e^{η_{2}})$ . The term $1$ is negligible. So $u \approx 2 κ_{2}^{2} sech^{2} (η_{2} /2 + \frac{1}{2} lo g A_{12})$ .

The phase shifts are:

$Δ x_{1} = \frac{1}{2 κ _{1}} lo g \frac{( κ _{1} + κ _{2} ) ^{2}}{( κ _{1} - κ _{2} ) ^{2}} > 0$ (faster soliton shifted forward).

$Δ x_{2} = \frac{1}{2 κ _{2}} lo g \frac{( κ _{1} - κ _{2} ) ^{2}}{( κ _{1} + κ _{2} ) ^{2}} < 0$ (slower soliton shifted backward).

Exercise 7 (hard, symbolic).

The nonlinear Schrodinger equation $i ψ_{t} + ψ_{xx} + 2∣ ψ ∣^{2} ψ = 0$ has a Lax pair. The Lax operator is the $2 \times 2$ system $L = i (\partial / \partial x \overset{ˉ}{ψ} - ψ - \partial / \partial x)$ . Show that the (1,2) entry of $d L / d t = [M, L]$ yields the NLS equation for an appropriate partner $M$ .

Hint

The partner is $M = (- 2 i \partial^{2} / \partial x^{2} + i ∣ ψ ∣^{2} - 2 \overset{ˉ}{ψ} \partial / \partial x - \overset{ˉ}{ψ}_{x} 2 ψ \partial / \partial x + ψ_{x} 2 i \partial^{2} / \partial x^{2} - i ∣ ψ ∣^{2})$ . Compute the (1,2) entry of $M L - L M$ .

Answer

The (1,2) entry of $d L / d t$ is $- i ψ_{t}$ (from differentiating $- ψ$ in $L$ ). Computing the (1,2) entry of $[M, L]$ :

$(M L)_{12} = M_{11} L_{12} + M_{12} L_{22} = (- 2 i D^{2} + i ∣ ψ ∣^{2}) (- ψ) + (2 ψ D + ψ_{x}) (- D) = 2 i D^{2} ψ - i ∣ ψ ∣^{2} ψ - 2 ψ D^{2} - ψ_{x} D$ .

$(L M)_{12} = L_{11} M_{12} + L_{12} M_{22} = i D (2 ψ D + ψ_{x}) + (- ψ) (2 i D^{2} - i ∣ ψ ∣^{2}) = 2 i ψ D^{2} + 2 i ψ_{x} D + i ψ_{xx} - 2 i ψ D^{2} + i ∣ ψ ∣^{2} ψ$ .

After subtracting and simplifying (using product rules for $D$ acting on $ψ$ ), the $D$ and $D^{2}$ terms cancel, leaving the zeroth-order term:

$[M, L]_{12} = i (ψ_{xx} + 2∣ ψ ∣^{2} ψ)$ .

Setting $- i ψ_{t} = i (ψ_{xx} + 2∣ ψ ∣^{2} ψ)$ gives $i ψ_{t} + ψ_{xx} + 2∣ ψ ∣^{2} ψ = 0$ , the focusing NLS equation.

Lean formalization Intermediate+

lean_status: none. Mathlib has no formalisation of the inverse scattering transform, Lax pairs, or soliton theory. The prerequisites that exist include linear operators on Hilbert spaces, spectral theory for compact self-adjoint operators, and ODE existence theory. What is entirely absent:

The Lax isospectral condition. Defining $d L / d t = [M, L]$ as an ODE on a space of operators and proving it preserves eigenvalues would require a theory of operator-valued ODEs, which is not in Mathlib.
Spectral theory of Schrodinger operators on the line. The direct scattering map (potential to scattering data) and the inverse scattering map (scattering data to potential) require the Jost function, the continuous spectrum, and the Gelfand-Levitan-Marchenko integral equation.
The N-soliton formula. The determinant formula $u = - 2 (d^{2} / d x^{2}) lo g det A$ and its verification require PDE analysis and matrix differential calculus.
Bi-Hamiltonian structures. The two compatible Poisson brackets for KdV, the Lenard recursion, and the proof that it generates infinitely many commuting conserved quantities require infinite-dimensional Poisson geometry.
The IST as a canonical transformation. Proving the scattering map is symplectic requires the symplectic structure on the space of scattering data.

The most accessible formalization target would be Exercise 2 (conservation of $tr (L^{k})$ from the Lax equation), which requires only finite-dimensional linear algebra and the cyclic property of the trace.

Advanced results Master

Riemann-Hilbert problems

The inverse scattering transform can be reformulated as a Riemann-Hilbert factorisation problem, which provides both a powerful analytical framework and a route to asymptotic analysis ^{[source pending]}.

A Riemann-Hilbert problem (RHP) asks: given a jump matrix $v (λ)$ defined on a contour $Γ$ in the complex $λ$ -plane, find a matrix-valued function $m (λ)$ that is analytic in $C ∖ Γ$ , has boundary values $m_{+} (λ)$ (from the left) and $m_{-} (λ)$ (from the right) on $Γ$ satisfying $m_{+} (λ) = m_{-} (λ) v (λ)$ , and normalises to the identity as $λ \to \infty$ .

For the KdV equation, the RHP is formulated on the continuous spectrum $λ > 0$ (the positive real axis) and at the discrete eigenvalues $λ = - κ_{n}^{2}$ . The jump matrix $v (λ)$ is constructed from the scattering data $r (k, t)$ , and the solution $u (x, t)$ is recovered from the expansion of $m (λ)$ at infinity.

The RHP formulation has two advantages over the Gelfand-Levitan-Marchenko equation. First, it generalises naturally to matrix equations (NLS, sine-Gordon, Toda lattice) where the scalar scattering problem is replaced by a matrix spectral problem. Second, it provides a systematic method for computing the long-time asymptotics of the solution via the Deift-Zhou steepest-descent method: deform the contour $Γ$ to make the jump matrix $v (λ)$ approach the identity on most of the contour, concentrating the singularity near special points (stationary-phase points). The leading-order contribution comes from these points and gives the asymptotic soliton positions and dispersive tails ^{[source pending]}.

IST for the nonlinear Schrodinger equation

The focusing nonlinear Schrodinger equation (NLS)

i ψ_{t} + ψ_{xx} + 2∣ ψ ∣^{2} ψ = 0

is the second major integrable PDE, discovered to be integrable by Zakharov and Shabat (1972) ^{[source pending]}. It models envelope dynamics of a carrier wave in a weakly nonlinear dispersive medium: the propagation of light pulses in optical fibres, Langmuir waves in plasmas, and water-wave envelopes.

The Zakharov-Shabat (ZS) spectral problem is the $2 \times 2$ system

\frac{d}{d x} (ϕ_{1} ϕ_{2}) = (- iλ - \overset{ˉ}{ψ} ψ iλ) (ϕ_{1} ϕ_{2}),

and the Lax pair is $d Φ/ d t = M Φ$ where $M$ is a $2 \times 2$ matrix differential operator in $x$ depending on $ψ$ , $\overset{ˉ}{ψ}$ , and $λ$ . The compatibility condition produces the NLS equation.

The IST for NLS follows the same three-step pattern as KdV: direct scattering (solve the ZS problem at $t = 0$ ), time evolution of scattering data (the discrete eigenvalues $λ_{n}$ are constant, the norming constants evolve by $e^{2 i λ_{n}^{2} t}$ ), and inverse scattering via a Riemann-Hilbert problem.

The NLS soliton is $ψ (x, t) = 2 η sech (2 η (x - 4 ξ t - x_{0})) e^{- 2 i ξ x - 4 i (ξ^{2} - η^{2}) t + i ϕ_{0}}$ , where $λ = ξ + i η$ is a discrete eigenvalue in the upper half-plane. The modulus $∣ ψ ∣$ is a sech-shaped pulse travelling at speed $4 ξ$ , with amplitude $2 η$ and width $1/ (2 η)$ .

The focusing NLS admits a phenomenon with no KdV analogue: modulational instability (Benjamin-Feir instability). The plane-wave solution $ψ = a e^{2 i a^{2} t}$ is unstable to long-wavelength perturbations, which grow exponentially. In the IST framework, this corresponds to complex eigenvalues in the ZS spectral problem emerging from the continuous spectrum via bifurcation. This mechanism underlies the formation of rogue waves in oceanography and optical fibres ^{[source pending]}.

Finite-gap solutions and algebraic geometry

The periodic KdV equation ( $u (x + L, t) = u (x, t)$ ) has a spectral theory that connects to algebraic geometry. The Schrodinger operator $L = - \partial^{2} / \partial x^{2} + u (x, t)$ with periodic potential is a Hill operator, and its spectrum consists of bands on the real axis separated by gaps. A generic potential has infinitely many open gaps, but a potential with finitely many gaps corresponds to a finite-dimensional integrable system ^{[source pending]}.

Novikov's conjecture (1974) states that a potential $u (x)$ is a finite-gap potential (the Hill operator has exactly $g$ open gaps) if and only if $u$ satisfies an ODE of the form $grad I_{g + 1} = const \cdot grad I_{1}$ for one of the higher KdV conserved quantities. These finite-gap potentials are the periodic analogues of the $N$ -soliton solutions.

Its and Matveev (1975) constructed the explicit finite-gap solutions using Riemann theta functions on hyperelliptic Riemann surfaces ^{[source pending]}. For $g$ gaps, the spectral curve

μ^{2} = j = 1 \prod 2 g + 1 (λ - E_{j})

is a hyperelliptic Riemann surface of genus $g$ , where $E_{1} < E_{2} < \dots < E_{2 g + 1}$ are the band edges. The solution is

u (x, t) = const - 2 \frac{d ^{2}}{d x ^{2}} lo g θ (U x + V t + Z),

where $θ$ is the Riemann theta function of the spectral curve, $U$ and $V$ are period vectors, and $Z$ is a phase vector. The $g$ -gap solution is a quasi-periodic function of $x$ with $g$ independent frequencies, living on a $g$ -dimensional invariant torus. The $g$ open gaps are the $g$ "action variables," and the phases on the Jacobian are the "angle variables."

The finite-gap theory connects the KdV equation to algebraic geometry: the phase space of $g$ -gap potentials is the Jacobian of the spectral curve, the KdV flow is a straight-line flow on the Jacobian, and the conserved quantities are the branch points $E_{j}$ . This is the same geometric framework that appears in the periodic Toda lattice 09.06.03.

Tau functions

The $N$ -soliton formula $u = - 2 (d^{2} / d x^{2}) lo g det A$ has a deeper algebraic origin in the theory of tau functions. Hirota (1971) introduced the bilinear substitution $u = - 2 (d^{2} / d x^{2}) lo g τ$ and showed that the KdV equation transforms into a bilinear equation for $τ$ ^{[source pending]}:

D_{x} (D_{t} + D_{x}^{3}) τ \cdot τ = 0,

where $D_{x}$ and $D_{t}$ are Hirota's bilinear operators: $D_{x} f \cdot g = (\partial / \partial x_{1} - \partial / \partial x_{2}) f (x_{1}) g (x_{2})_{x_{2} = x_{1}}$ .

The tau function has a remarkable algebraic interpretation discovered by Sato (1981): the space of solutions of integrable equations is identified with an orbit of the infinite-dimensional group $GL (\infty)$ acting on the Fermionic Fock space, and the tau function is a certain determinant on this space ^{[source pending]}. Sato showed that all integrable equations in the KP hierarchy are encoded in a single bilinear equation for the tau function, and the individual equations (KdV, Boussinesq, sine-Gordon) are obtained by specialising the independent variables.

For the $N$ -soliton solution, the tau function is the Wronskian determinant of $N$ exponential functions, which is exactly $det A$ . For finite-gap solutions, the tau function is the Riemann theta function of the spectral curve. The tau function unifies the soliton and finite-gap pictures: both are special evaluations of the same algebraic object.

The Hirota bilinear method is also a practical tool for constructing multi-soliton solutions. By making the ansatz $τ = 1 + ϵ e^{η_{1}} + ϵ e^{η_{2}} + ϵ^{2} A_{12} e^{η_{1} + η_{2}}$ and substituting into the bilinear equation, one reads off the phase factor $A_{12} = (κ_{1} - κ_{2})^{2} / (κ_{1} + κ_{2})^{2}$ without solving any differential equation ^{[source pending]}.

Synthesis

The Lax pair and inverse scattering transform constitute the infinite-dimensional generalisation of the Liouville-Arnold theorem 09.06.02. The eigenvalues of the Lax operator are the action variables, the scattering data provides the angle variables, and the isospectral condition is the involution of the conserved quantities. The passage from finite-dimensional integrable systems to integrable PDEs is not an analogy but a mathematical identification: the Toda lattice 09.06.03 interpolates between the two, and the KdV equation is its continuum limit.

The Riemann-Hilbert formulation provides the analytical framework for asymptotic analysis. The finite-gap theory connects to algebraic geometry through spectral curves and Jacobians. The tau function unifies the soliton and finite-gap pictures through Sato theory. Together, these three perspectives -- analytical (IST/RHP), geometrical (finite-gap/Jacobians), and algebraic (tau functions/Sato theory) -- provide a complete description of the integrable PDE.

Historical context Master

Russell and the canal wave (1834). John Scott Russell's observation of a solitary wave on the Union Canal near Edinburgh was the first recorded encounter with a soliton. Russell spent the next decade conducting tank experiments, measuring the wave's speed and arguing for its physical reality against Airy and Stokes, who believed all localised water waves must disperse. Korteweg and de Vries (1895) derived the KdV equation as a long-wave approximation to the full Euler equations, resolving the dispute in Russell's favour ^{[source pending]}.

The FPU experiment (1955). Enrico Fermi, John Pasta, Stanislaw Ulam, and Mary Tsingou simulated a nonlinear lattice on the MANIAC computer at Los Alamos, expecting equipartition of energy among normal modes. Instead, they found that energy fed into the lowest mode returned almost entirely to that mode after a recurrence time. This "FPU recurrence" was unexplained until Kruskal and Zabusky showed that the continuum limit of the FPU lattice is the KdV equation and that the recurrence is caused by soliton collisions ^{[source pending]}.

Zabusky and Kruskal (1965). Zabusky and Kruskal simulated the KdV equation numerically and discovered that an initial cosine wave broke up into a train of solitons of different heights, which passed through each other and re-formed the original profile. They coined the word "soliton" to emphasise the particle-like collision properties. Their paper is one of the foundational works of computational physics ^{[source pending]}.

Gardner, Greene, Kruskal, and Miura (1967). The GGKM team discovered that the KdV equation can be solved by the inverse scattering method: the Schrodinger operator with potential $u (x, t)$ has constant eigenvalues under the KdV flow, and the solution can be reconstructed from the time-evolved scattering data via the Gelfand-Levitan-Marchenko equation. This was the first instance of the inverse scattering transform ^{[source pending]}.

Lax (1968). Peter Lax reformulated the GGKM discovery as the Lax pair equation $d L / d t = [M, L]$ , providing the algebraic framework that has become the standard language of integrable systems. The Lax formulation made it possible to identify and solve other integrable equations by finding their Lax pairs ^{[source pending]}.

Zakharov and Shabat (1972). Extended the IST to the nonlinear Schrodinger equation using a $2 \times 2$ matrix spectral problem, demonstrating that the method was not specific to KdV. This was followed by IST solutions for sine-Gordon (Ablowitz, Kaup, Newell, and Segur, 1973) and many other equations ^{[source pending]}.

Hirota (1971), Sato (1981), and the algebraic theory. Hirota's bilinear method provided a systematic way to construct multi-soliton solutions. Sato's identification of tau functions with orbits on the Fermionic Fock space placed the entire theory in a unified algebraic framework. The finite-gap theory of Novikov, Its, Matveev, and others (1974-1975) connected the periodic problem to algebraic geometry. The $r$ -matrix formalism of Faddeev, Takhtajan, and Semenov-Tian-Shansky (late 1970s-1980s) provided the Lie-algebraic explanation for why the Lax structure produces commuting conserved quantities ^{[source pending]}.

Connections Master

Liouville-Arnold theorem 09.06.02 -- the Lax pair and IST provide the infinite-dimensional realisation of the Liouville-Arnold framework. The eigenvalues of $L$ are the action variables, the scattering data provides the angle variables, and the isospectral condition is the involution of the conserved quantities.
Action-angle variables 09.06.01 -- the IST is the infinite-dimensional action-angle map. The direct scattering transform plays the role of the coordinate change to action-angle variables, and the inverse scattering transform is its inverse.
Completely integrable systems: Toda lattice 09.06.03 -- the Toda lattice Lax pair is the finite-dimensional prototype of the KdV Lax pair. The Toda lattice is the discrete analogue of KdV, and the continuum limit of Toda produces KdV. Both share the isospectral deformation structure.
KAM theorem 09.08.01 -- the Lax pair and IST provide the exact solution of the integrable PDE. KAM theory studies which structures survive when the integrable system is perturbed; the infinite hierarchy of conserved quantities from the Lax structure is the starting point for perturbation analysis.
Hamiltonian formalism 09.04.01 -- the bi-Hamiltonian structure of KdV (two compatible Poisson brackets) is an infinite-dimensional generalisation of the finite-dimensional Hamiltonian formalism. The Lenard recursion between the two brackets generates the infinite hierarchy of commuting Hamiltonians.
Dynamics spine [38.07] -- the IST for KdV is the paradigmatic example of the inverse scattering method in the dynamics spine. The spectral theory of Schrodinger operators connects to the broader spectral and scattering theory framework.
Numerical PDE methods 49.03.01 -- numerical methods for KdV must preserve the isospectral structure to avoid spurious radiation. The Lax pair distinguishes integrable PDEs from generic dispersive equations and motivates geometric integrators.
Algebraic geometry -- the finite-gap theory for periodic KdV connects Hamiltonian mechanics to the theory of Riemann surfaces, Jacobians, and theta functions. This is the same geometric framework that appears in the periodic Toda lattice.
Quantum mechanics -- the Schrodinger operator $L = - \partial^{2} / \partial x^{2} + u$ is the one-dimensional Schrodinger operator from quantum mechanics. The IST exploits the spectral theory of this operator to solve a classical PDE, creating a bridge between quantum scattering theory and classical nonlinear waves.

Bibliography Master

Zabusky, N. J. and Kruskal, M. D. "Interaction of 'solitons' in a collisionless plasma." Phys. Rev. Lett. 15 (1965), 240--243. ^{[source pending]}
Gardner, C. S., Greene, J. M., Kruskal, M. D., and Miura, R. M. "Method for solving the Korteweg-de Vries equation." Phys. Rev. Lett. 19 (1967), 1095--1097. ^{[source pending]}
Lax, P. D. "Integrals of nonlinear equations of evolution and solitary waves." Comm. Pure Appl. Math. 21 (1968), 467--490. ^{[source pending]}
Zakharov, V. E. and Shabat, A. B. "Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media." Soviet Phys. JETP 34 (1972), 62--69. ^{[source pending]}
Hirota, R. "Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons." Phys. Rev. Lett. 27 (1971), 1192--1194. ^{[source pending]}
Novikov, S. P. "The periodic problem for the Korteweg-de Vries equation." Funct. Anal. Appl. 8 (1974), 236--246. ^{[source pending]}
Its, A. R. and Matveev, V. B. "Hill's operator with infinitely many zones." Funct. Anal. Appl. 9 (1975), 65--66. ^{[source pending]}
Sato, M. "Soliton equations as dynamical systems on infinite-dimensional Grassmann manifold." RIMS Kokyuroku 439 (1981), 30--46. ^{[source pending]}
Date, E., Jimbo, M., Kashiwara, M., and Miwa, T. "Transformation groups for soliton equations." J. Phys. Soc. Japan 50 (1981), 3806--3812. ^{[source pending]}
Faddeev, L. D. and Takhtajan, L. A. Hamiltonian Methods in the Theory of Solitons. Springer, 2007. ^{[source pending]}
Drazin, P. G. and Johnson, R. S. Solitons: An Introduction. Cambridge, 1989. ^{[source pending]}
Ablowitz, M. J. and Segur, H. Solitons and the Inverse Scattering Transform. SIAM, 1981. ^{[source pending]}
Novikov, S. P., Manakov, S. V., Pitaevskii, L. P., and Zakharov, V. E. Theory of Solitons: The Inverse Scattering Method. Plenum, 1984. ^{[source pending]}
Ablowitz, M. J. and Clarkson, P. A. Solitons, Nonlinear Evolution Equations and Inverse Scattering. LMS Lecture Notes 149, Cambridge, 1991. ^{[source pending]}
Deift, P. and Zhou, X. "A steepest descent method for oscillatory Riemann-Hilbert problems." Bull. Amer. Math. Soc. 26 (1992), 119--123. ^{[source pending]}
Goldstein, H., Poole, C., and Safko, J. Classical Mechanics, 3rd ed. Pearson, 2002. ^{[source pending]}
Landau, L. D. and Lifshitz, E. M. Mechanics, 3rd ed. Pergamon, 1976. ^{[source pending]}

Prerequisites

09.06.02 pending
09.06.03 pending

Used in

09.08.01

Tier anchors

beginner: Drazin & Johnson, Solitons: An Introduction (1989), Ch. 1-2; Zabusky & Kruskal, Phys. Rev. Lett. 15 (1965)
intermediate: Ablowitz & Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering (1991), Ch. 1-3; Lax, Comm. Pure Appl. Math. 21 (1968)
master: Faddeev & Takhtajan, Hamiltonian Methods in the Theory of Solitons (2007); Novikov et al., Theory of Solitons (1984); Ablowitz & Segur, Solitons and the Inverse Scattering Transform (1981)

References

Goldstein, Poole & Safko — Classical Mechanics, 3rd ed. (Pearson, 2002) · Ch. 3 The central-force problem; connection between symmetries and conserved quantities
Landau & Lifshitz — Mechanics, 3rd ed. (Pergamon, 1976) · §49 Small oscillations of coupled systems
Zabusky & Kruskal — Phys. Rev. Lett. 15 (1965) · Interaction of 'solitons' in a collisionless plasma
Gardner, Greene, Kruskal & Miura — Phys. Rev. Lett. 19 (1967) · Method for solving the Korteweg-de Vries equation
Lax — Comm. Pure Appl. Math. 21 (1968) · Integrals of nonlinear equations of evolution and solitary waves
Drazin & Johnson — Solitons: An Introduction (Cambridge, 1989) · Ch. 1 The KdV equation; Ch. 2 The inverse scattering transform
Ablowitz & Clarkson — Solitons, Nonlinear Evolution Equations and Inverse Scattering (LMS 149, 1991) · Ch. 1-3 KdV, Lax pairs, NLS; Riemann-Hilbert problems
Faddeev & Takhtajan — Hamiltonian Methods in the Theory of Solitons (Springer, 2007) · Part I The inverse scattering method; Ch. 1 The Lax representation
Novikov, Manakov, Pitaevskii & Zakharov — Theory of Solitons (Plenum, 1984) · Ch. II The inverse scattering method; Ch. III Finite-gap theory
Zakharov & Shabat — Soviet Phys. JETP 34 (1972) · Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media
Ablowitz & Segur — Solitons and the Inverse Scattering Transform (SIAM, 1981) · Ch. 2 IST for KdV; Ch. 3 IST for NLS
Its & Matveev — Functional Anal. Appl. 9 (1975) · Hill's operator with infinitely many zones
Hirota — Phys. Rev. Lett. 27 (1971) · Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons
Sato — RIMS Kokyuroku 439 (1981) · Soliton equations as dynamical systems on infinite-dimensional Grassmann manifold
Date, Jimbo, Kashiwara & Miwa — J. Phys. Soc. Japan 50 (1981) · Transformation groups for soliton equations

Estimated time

beginner: 20m
intermediate: 50m
master: 70m