Korteweg–de Vries equation

The Korteweg–de Vries equation describes a balance between nonlinear transport and dispersion. In the normalization used here, a positive localized pulse of height $2\kappa^2$ travels at speed $4\kappa^2$ without changing shape. This record fixes the real-line, zero-background model and identifies the spectral structure behind that solution. The KdV learning sequence develops the travelling wave, conservation laws, scattering problem, and reconstruction as separate calculations.

Required background. Differentiate a function of two variables and integrate by parts. Helpful background. The nonlinear-wave conventions translate the signs and normalizations used in the spectral problem.

KdV on the real line

Let $u(x,t)$ be a real field with $x\in\mathbb R$. The equation and initial data are

$$u_t+6uu_x+u_{xxx}=0, \qquad u(x,0)=u_0(x).$$

For the calculations below, assume a smooth solution on the time interval considered, with $u(\cdot,t)$ and its spatial derivatives rapidly decreasing at both ends of the line. Schwartz functions provide a convenient class: every spatial derivative decays faster than every inverse power of $|x|$. Assume enough uniform control on bounded time intervals to differentiate the integrals used below. These assumptions state the setting of the calculations; they are not a proof of existence for an arbitrary initial datum.

The field is normalized and need not be a dimensional surface height. Its interpretation depends on the physical derivation. KdV arose as an approximation for long water waves; Lax 1968, report PDF, p. 1, equation (1.1) uses the scaled field $w=6u$, so his equation is $w_t+ww_x+w_{xxx}=0$. Using the name “KdV” in a physical application does not establish the approximation’s error or range of validity.

Nonlinearity and dispersion

Without dispersion, $u_t+6uu_x=0$ transports a local field value at speed $6u$. Without the nonlinear term, substituting $u=e^{i(kx-\omega t)}$ into $u_t+u_{xxx}=0$ gives

$$\omega(k)=-k^3, \qquad v_{\mathrm{group}}=-3k^2.$$

Thus small linear wave packets move toward decreasing $x$ in this frame. The positive soliton below moves toward increasing $x$. There is no contradiction: its finite amplitude makes the nonlinear term essential.

The positive one-soliton

For $\kappa\gt0$ and $x_0\in\mathbb R$, define

$$u(x,t)=2\kappa^2\operatorname{sech}^2\!\left[\kappa(x-4\kappa^2t-x_0)\right].$$

Its peak is at $X(t)=4\kappa^2t+x_0$, its height is $2\kappa^2$, and its characteristic width is $\kappa^{-1}$. A taller member of this family is narrower and faster. The derivation follows the travelling-wave reduction in Lax 1968, report PDF, pp. 2–3, equations (1.5)–(1.8), with $w=6u$.

To check the formula directly, write $u=U(\xi)$, $\xi=x-vt-x_0$. Decay fixes the integration constant in the equation to zero:

$$U''=vU-3U^2.$$

Multiplication by $U'$ and a second integration give

$$(U')^2=vU^2-2U^3.$$

For the proposed profile, $U''=4\kappa^2U-3U^2$. Thus $v=4\kappa^2$ and the original PDE follows by differentiating the second-order equation. This verifies an exact solution. It does not show that every localized initial profile is a soliton, or that a perturbed pulse remains close to one.

The travelling-wave lesson supplies the integration steps and a nonzero-background comparison. The one-soliton reconstruction obtains the same field from a negative eigenvalue and a positive normalization coefficient.

Three conserved quantities

All integrals in this section use Lebesgue measure $dx$ over $\mathbb R$. Define

$$\begin{aligned} M[u]&=\int_{\mathbb R}u\,dx,\\ P[u]&=\int_{\mathbb R}u^2\,dx,\\ E[u]&=\int_{\mathbb R}\left(u^3-\frac12u_x^2\right)dx. \end{aligned}$$

These are the $M$, $P$, and $E$ used throughout the learning sequence. The conventional momentum functional is often $P/2$; our Hamiltonian below is $H=-E$. Such names and constant factors do not by themselves identify the conserved mass or energy of a parent fluid model. Neither $E$ nor $H$ is positive definite for general fields.

The equation is a local conservation law,

$$u_t+\partial_x(3u^2+u_{xx})=0,$$

so $\dot M=0$ when its flux vanishes at both infinities. Similarly,

$$\partial_t(u^2) +\partial_x\!\left(4u^3+2uu_{xx}-u_x^2\right)=0,$$

which gives $\dot P=0$. The conservation lesson derives these fluxes and checks the endpoint terms.

For the third quantity, let $h=\delta E/\delta u=u_{xx}+3u^2$. Since $u_t=-\partial_xh$, integration by parts gives

$$\frac{dE}{dt}=\int_{\mathbb R}h\,u_t\,dx =-\int_{\mathbb R}h\,\partial_xh\,dx=0.$$

No division by $u$ or restriction to a travelling wave entered these arguments. They therefore apply to every solution with the stated smoothness and decay, not just the explicit pulse. The quantities $P$ and $E$ correspond to the first two integrals discussed in Lax 1968, report PDF, § 2, pp. 24–26, equations (2.7)–(2.9): with $w=6u$, his $I_1=18P$ and $I_2=72E$.

A quantitative soliton check

For the one-soliton, substitution of $z=\kappa(x-X)$ gives

$$M=4\kappa, \qquad P=\frac{16}{3}\kappa^3, \qquad E=\frac{32}{5}\kappa^5.$$

For example,

$$\int_{\mathbb R}u_x^2\,dx=\frac{64}{15}\kappa^5, \qquad \int_{\mathbb R}u^3\,dx=\frac{128}{15}\kappa^5.$$

The Hamiltonian $H=-E$ therefore has the negative value $-32\kappa^5/5$. Replacing a Hamiltonian density by its negative requires changing the Poisson operator as well if the same PDE is to result. These closed values are useful for numerical refinement tests; small drift of three integrals alone does not establish that an entire numerical solution is accurate.

Hamiltonian formulation

On functionals for which the following variational derivatives and integrations by parts are defined, use

$$\{F,G\}=\int_{\mathbb R} \frac{\delta F}{\delta u}\, \partial_x\!\left(\frac{\delta G}{\delta u}\right)dx.$$

The constant-coefficient Poisson operator is $J=\partial_x$. Define the Hamiltonian

$$H[u]=-E[u]=\int_{\mathbb R}\left(\frac12u_x^2-u^3\right)dx.$$

Then

$$u_t=J\frac{\delta H}{\delta u} =\partial_x(-u_{xx}-3u^2).$$

Integration by parts makes the bracket antisymmetric; its constant Poisson operator gives the formal Jacobi identity. This is a field-theoretic Hamiltonian description, not a finite-dimensional count of independent Liouville integrals. The mass is a Casimir for this bracket, since $\delta M/\delta u=1$ is annihilated by $\partial_x$. The functional $P/2$ generates the infinitesimal shift $u_s=u_x$; $P$ itself generates $u_s=2u_x$.

Spectral structure and its scope

With $D=\partial_x$, take

$$L=-D^2-u, \qquad A=-4D^3-6uD-3u_x.$$

On smooth test functions the differential expressions satisfy

$$[A,L]=(u_{xxx}+6uu_x),$$

where the right side acts by multiplication. Therefore the equation $L_t=[A,L]$ is precisely KdV. For real smooth bounded $u$, the Schrödinger operator $L$ is self-adjoint on $H^2(\mathbb R)\subset L^2(\mathbb R,dx)$. The commutator calculation does not by itself establish the domain and evolution statements needed to promote a formal differential identity into unitary equivalence of unbounded operators. The Library derivation separates these issues and verifies the one-soliton spectral problem explicitly.

The pulse corresponds to the attractive potential $-u$ with one bound-state eigenvalue $-\kappa^2$ and zero reflection for real nonzero wave number. Its eigenvalue fixes the amplitude and speed; a norming coefficient fixes its position. The scattering-data normalization is specified in Grunert and Teschl 2009, author PDF, § 2, pp. 4–5, equations (2.3)–(2.13), whose potential is $q=-u$.

This record establishes the Lax identity, three conservation laws, and a checked reflectionless solution. The full inverse-scattering method requires the admissible scattering data, an inverse problem, and analytic existence and uniqueness results. A formal Lax pair alone does not provide those results.

The two-soliton derivation adds an exact interacting example with two distinct negative eigenvalues. Its asymptotic pulses retain their amplitudes and speeds but shift position. The collision laboratory compares numerical evolution with that complete solution; an independently translated sum of two isolated pulses is not the exact interaction.

Scaling and other boundary regimes

If $u(x,t)$ solves the normalized equation, then for $a\gt0$

$$u_a(x,t)=a^2u(ax,a^3t)$$

also solves it. Every term gains the same factor $a^5$. In dimensional bookkeeping compatible with the normalized coefficients, $[t]=[x]^3$ and $[u]=[x]^{-2}$. These are scaling dimensions of this equation, not an identification of physical time with a cubed length.

More generally, after removing a constant advection speed by a moving frame, consider

$$U_T+\alpha UU_X+\beta U_{XXX}=0.$$

For $\alpha\ne0$, $\beta\gt0$, and a chosen length $\ell\gt0$, the change

$$X=\ell x, \qquad T=\frac{\ell^3}{\beta}t, \qquad U=\frac{6\beta}{\alpha\ell^2}u$$

gives the normalized model. The sign of $\alpha$ then determines whether the positive normalized soliton is an elevation or a depression in $U$. If a coefficient vanishes or has a different sign, redo the scaling rather than silently using this conversion.

The same PDE also supports different spectral problems:

Domain and asymptotics	What changes
Real line, $u\to0$ at both ends	The scattering and localized soliton calculations on this page apply.
Periodic interval	Periodic or Floquet spectral data replace the two-end Jost normalization. A cut-off line soliton is not an exact periodic solution.
Same nonzero limit $u\to b$ at both ends	Subtracting the background also shifts the moving frame; the unrenormalized integrals above generally diverge.
Different limits at the two ends	There is a step-background scattering problem; the zero-background reconstruction cannot simply be reused.
Half-line or finite interval	Boundary conditions and their time evolution matter. Boundary fluxes need not vanish, and a bulk Lax identity does not establish an integrable boundary problem.

For a common background $b$, the explicit conversion is $u(x,t)=b+v(x-6bt,t)$. Substitution shows that $v$ satisfies the same normalized KdV equation. The auxiliary Schrödinger continuum is shifted by $-b$; the original zero-background spectrum and integrals must be interpreted accordingly.

References

• Grunert, Katrin, and Gerald Teschl. “Long-time asymptotics for the Korteweg–de Vries equation via nonlinear steepest descent.” Mathematical Physics, Analysis and Geometry 12 (2009), 287–324. DOI. Open PDF. Locators above use the author PDF’s printed pages.
• Lax, Peter D. “Integrals of nonlinear equations of evolution and solitary waves.” Communications on Pure and Applied Mathematics 21 (1968), 467–490. DOI. Open report PDF, NYO-1480-87, January 1968. Locators above use the report’s printed pages, not the journal pagination.