Verify KdV conservation laws

What stays fixed while a KdV wave changes shape? Three useful answers are the integral of the field, the integral of its square, and a cubic-gradient combination. You will derive their conservation by integration by parts, evaluate all three on a one-soliton solution, and identify the boundary flux that invalidates a careless finite-interval argument. These checks constrain a solution; they do not uniquely determine it.

Required background. Use the profile and derivative identity from Derive a KdV travelling wave. You need the product rule and integration by parts; the entry check below makes the endpoint term explicit.

Helpful background. The convention reference distinguishes normalized invariants from physical mass, momentum or energy in a particular application.

Three integrals of the decaying line problem

Let $u(x,t)$ be a real smooth solution of

$$u_t+6u u_x+u_{xxx}=0,\qquad x\in\mathbb R.$$

Assume rapid spatial decay of $u$, the derivatives used below, and sufficient uniform control on a finite time interval to differentiate under the integrals. Schwartz-class spatial profiles with smooth controlled time dependence are a sufficient setting for these calculations. Define

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

The names $M,P,E$ are labels for these precise formulas; factors and signs vary across sources. In particular $E$ is not a positive norm. The cubic-gradient conserved functional appears in Lax 1968, report pp. 24–26, equations (2.7)–(2.9), PDF. With his field $w=6u$, his $I_2(w)=\int(w^3/3-w_x^2)dx$ equals $72E$.

Entry check and repair

Is $\int_a^b u u_{xxx}\,dx$ zero for every smooth $u$?

Repair. One integration by parts followed by the product rule gives

$$\int_a^b u u_{xxx}\,dx =\left[u u_{xx}-\frac12u_x^2\right]_a^b.$$

The bracket means the value at $b$ minus the value at $a$. It vanishes for the decaying line limit, or matching periodic derivatives over a full period. Smoothness on a finite interval alone does not make it vanish.

Conservation from fluxes

Write the PDE as

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

Integrating over $[a,b]$ gives

$$\frac{d}{dt}\int_a^b u\,dx =-\left[3u^2+u_{xx}\right]_a^b.$$

Sending both endpoints to infinity under the stated decay assumptions proves $\dot M=0$.

For the quadratic integral, use the entry-check identity:

$$\begin{aligned} \frac{d}{dt}\int_a^b u^2\,dx &=-12\int_a^b u^2u_x\,dx-2\int_a^b uu_{xxx}\,dx\\ &=-\left[4u^3+2u u_{xx}-u_x^2\right]_a^b. \end{aligned}$$

Again the full-line boundary terms vanish, proving $\dot P=0$. This derivation also supplies the flux you must retain when a pulse enters or leaves a numerical observation window.

The third integral is shorter if you first calculate its variation. Put

$$G=3u^2+u_{xx}.$$

Since $u_t=-G_x$, differentiating $E$ and integrating its derivative term by parts yields

$$\begin{aligned} \dot E &=\int_{\mathbb R}(3u^2u_t-u_xu_{xt})\,dx\\ &=\int_{\mathbb R}(3u^2+u_{xx})u_t\,dx\\ &=-\int_{\mathbb R}GG_x\,dx =-\frac12[G^2]_{-\infty}^{\infty}=0. \end{aligned}$$

At the second equality, the omitted term is $-[u_xu_t]_{-\infty}^{\infty}$, which also vanishes. The argument depends on the whole expression $G$; dropping either its nonlinear or derivative term destroys the cancellation.

Evaluate the invariants on one soliton

For $u=2\kappa^2\operatorname{sech}^2 z$, where $z=\kappa(x-4\kappa^2t-x_0)$, use $dx=dz/\kappa$ and $y=\tanh z$. Then $dy=\operatorname{sech}^2z\,dz$, and

$$\begin{aligned} \int_{\mathbb R}\operatorname{sech}^2z\,dz&=2,\\ \int_{\mathbb R}\operatorname{sech}^4z\,dz&=\int_{-1}^{1}(1-y^2)\,dy=\frac43,\\ \int_{\mathbb R}\operatorname{sech}^6z\,dz&=\int_{-1}^{1}(1-y^2)^2\,dy=\frac{16}{15}. \end{aligned}$$

Consequently

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

The first integral from the travelling-wave ODE gives $u_x^2=4\kappa^2u^2-2u^3$. Therefore

$$\begin{aligned} \int_{\mathbb R}u^3dx&=\frac{128}{15}\kappa^5,\\ \int_{\mathbb R}u_x^2dx&=\frac{64}{15}\kappa^5,\\ E&=\frac{32}{5}\kappa^5. \end{aligned}$$

For the worked pulse with $\kappa=1/2$, these become $M=2$, $P=2/3$ and $E=1/5$. None depends on $t$ or $x_0$. Their scaling dimensions, respectively $\ell^{-1},\ell^{-3},\ell^{-5}$, agree with the powers of $\kappa$.

What invariant checks can miss

A translated profile $U(x-wt)$ has the same values of every translation-invariant spatial integral for any speed $w$. The preceding lesson showed that the KdV residual is nonzero when $w\ne4\kappa^2$. Thus even perfect preservation of these three numbers cannot certify correct dynamics.

For numerical work, compare invariant drift with the error in the field itself, refine the discretization and track boundary effects independently. The convergence laboratory makes these distinctions executable. On a periodic domain the same integrations cancel between matched endpoints, but that is a periodic conservation statement, not evidence that a truncated pulse exactly solves the line problem.

Exercises

Guided practice: recover the missing gradient integral

For the soliton, use $u_x^2=4\kappa^2u^2-2u^3$ and the two stated integrals to derive $\int u_x^2dx$ and $E$. Check the numerical values for $\kappa=1$.

Hint

Keep a common denominator of $15$. The cubic part of $E$ is reduced by half the gradient integral.

Solution

Directly,

$$\int u_x^2dx =4\kappa^2\frac{16}{3}\kappa^3 -2\frac{128}{15}\kappa^5 =\frac{64}{15}\kappa^5.$$

Hence $E=(128/15-32/15)\kappa^5=32\kappa^5/5$. At $\kappa=1$ the gradient integral is $64/15$ and $E=32/5$. Both are positive for this pulse; the general functional $E$ need not be.

Independent practice: a centroid that sees the speed

Assume also enough weighted decay for $Q(t)=\int_{\mathbb R}xu(x,t)dx$ to exist. Prove $\dot Q=3P$. For a pulse with $M\ne0$, calculate the speed of its centroid $X=Q/M$ and compare it with the travelling-wave result.

Hint

Integrate $-\int x\partial_x(3u^2+u_{xx})dx$ by parts. The remaining integral of $u_{xx}$ is another boundary term.

Solution

The weighted boundary term vanishes by assumption, so

$$\dot Q=\int_{\mathbb R}(3u^2+u_{xx})dx=3P.$$

Since $\dot M=0$, $\dot X=3P/M$. For the soliton this equals

$$\dot X=\frac{3(16\kappa^3/3)}{4\kappa}=4\kappa^2.$$

Unlike unweighted invariant drift, this relation detects translating the pulse at an incorrect speed. For a sign-changing field with $M=0$ the centroid quotient is undefined, even though the identity for $Q$ can remain valid.

Transfer: observe a finite window

The pulse $U(x-vt)$ crosses a fixed window $[a,b]$. Show from its travelling-wave ODE that

$$\frac{d}{dt}\int_a^b u\,dx=v[u(a,t)-u(b,t)].$$

Does a changing window integral demonstrate failure of mass conservation?

Hint

The ODE says $3u^2+u_{xx}=vu$. Insert this in the finite-window flux formula.

Solution

The boundary formula gives $\dot M_{[a,b]}=-[vu]_a^b=v(u(a,t)-u(b,t))$. If a right-moving pulse is entering through $a$, its entering flux can exceed the flux leaving through $b$, increasing the window integral. Conservation on the entire line is fully compatible with this change. Calling it numerical drift without accounting for flux confuses the observation region with the full system.

From conserved integrals to spectral data

You have proved three conservation laws under explicit decay assumptions and obtained an independent speed check. Next, relate the wave to auxiliary scattering: a linear spectral problem will encode the soliton’s width in a bound-state eigenvalue.

References

• Lax, Peter D. Integrals of Nonlinear Equations of Evolution and Solitary Waves. Courant Institute report NYO-1480-87, January 1968. Open report PDF. Published version: Communications on Pure and Applied Mathematics 21, 467–490 (1968), DOI. Page and equation locators above refer to the report.