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Put $E_k=e^{i(kx+k^{3}t)}$, $I={\int }_L\phi (l)d\lambda (l)$ and $q=-I_x$. For the dressing operator $\mathcal{F}h=h+i{E}_k{\int }_{L}h(l)/(k+l)d\lambda (l)$, differentiating gives $\mathcal{F}(M\phi )=0$ and $\mathcal{F}(N\phi )=3ikM\phi$, where $M={\partial }_x^2-ik{\partial }_x+q$ and $N={\partial }_t+{\partial }_x^3+3q{\partial }_x$. Uniqueness of the linear integral equation therefore supplies a Lax pair. Its commutator identity is $[N,M]=-3q_{x}M+(q_t+q_{xxx}+6q{q}_x)$, which gives the KdV equation when the eigenfunction is not identically zero. Differentiation under the integral and the uniqueness assumption are essential hypotheses.

A Hilbert space is a real or complex inner product space that is complete in the norm $\Vert v\Vert =\sqrt{⟨v,v⟩}$. Completeness means that every Cauchy sequence has a limit in the space: if its terms become arbitrarily close in this norm, they converge to an allowed vector. The inner product supplies lengths and orthogonality; completeness ensures that limiting approximation procedures do not leave the space.

Finite-dimensional Euclidean spaces are examples, but the same geometry also works for sequences and functions. In the l2 sequence space,In the L2 space on $[-1,1]$ with Lebesgue measure,These formulas use the convention linear in the first argument. Elements of the L2 space are equivalence classes of square-integrable functions agreeing almost everywhere, so zero distance really identifies the same element. A vector can therefore be a whole function, and its squared length can measure an integrated squared error.

For example, approximating $f(x)=x^2$ by $a+bx$ is a nearest-point problem in this Hilbert space. The allowed approximations form the linear span of $1,x$, and the orthogonal projection is the constant $1/3$. Its residual $x^2-1/3$ is orthogonal to both allowed directions because its integral and its inner product with $x$ vanish. The geometry therefore identifies the best approximation without searching through every pair of coefficients.

This is useful whenever a complicated vector must be represented by a tractable collection of directions. Fourier series approximate functions with orthogonal oscillations; the Parseval identity relates the sum of squared coefficients to the squared norm. Least-squares solutions approximate attainable data using a linear operator. The Hilbert projection theorem explains why a nearest element exists and is unique for a nonempty closed convex set. Distinct application settings share the same geometry of useful components and orthogonal residuals. Casey Rodriguez's MIT lectures develop these connections between Hilbert spaces, Fourier series, and nearest-point problems.^[ref]

A historical mathematical setting is David Hilbert's Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen (1912). Its preface states that it reprints six communications published during 1904–1910 on linear integral equations. This provides a concrete source for the integral-equation setting associated with the name, while the abstract definition above incorporates later developments.^[ref]

A linear integral equation depends linearly on its unknown function. For example, $(I+K)f=g$ with $(Kf)(x)=\int K(x,y)f(y)dy$ has a trivial homogeneous null space whenever its inhomogeneous problem has a unique solution: otherwise a nonzero homogeneous solution could be added to that solution. This is the uniqueness step used in dressing constructions of a Lax pair.

Write $E_k=e^{i(kx+k^{3}t)}$, $I={\int }_L\phi (l)d\lambda (l)$ and $A_k={\int }_L\phi (l)/(k+l)d\lambda (l)$. Introduce the dressing operatorThe given linear integral equation is $\mathcal{F}\phi =E$, and $q=-I_x$. The appropriate contour and measure must permit differentiation under these integrals and avoid untreated kernel singularities.

For the specified spatial operator, direct differentiation givesInside the integral, replace ${\phi }_{xx}(l)+q\phi (l)$ by $M_l\phi (l)+il{\phi }_x(l)$. It follows thatApply $M_k$ to $\phi +i{E}_k{A}_k=E_k$. The last term contributes $-E_k{I}_x=q{E}_k$, canceling the right side, soThis is precisely the homogeneous integral equation. Uniqueness of the original inhomogeneous solution implies a trivial homogeneous null space, hence $M\phi =0$. This calculation is the spatial identity in Cauchy-kernel dressing for the KdV equation.

For the specified time operator, $N_k{E}_k=3ikq{E}_k$, because the derivatives of $e^{i(kx+k^{3}t)}$ cancel between ${\partial }_t$ and ${\partial }_x^3$. Expanding the derivatives of $E_k{A}_k$ givesFor completeness, the extra derivative terms before simplification are $3ik{A}_{k,xx}-3k^2{A}_{k,x}+3ikq{A}_k$. Substituting ${\phi }_{xx}+q\phi =M_l\phi +il{\phi }_x$ turns their derivative contribution into $-3k{\int }_L{\phi }_{x}d\lambda =-3k{I}_x$.

Applying $N_k$ to the original linear integral equation and using $I_x=-q$ cancels its right side. ThusPart (a) gives $E_k{\int }_L{M}_l\phi /(k+l)d\lambda =i{M}_k\phi$. Therefore the exact requested identity isSince $M\phi =0$, this becomes homogeneous too. The same uniqueness argument gives $N\phi =0$; it is not necessary to assume a separate uniqueness theorem for the differential time equation.

The two differentiated linear integral equations have now supplied both equations of the Lax pair. To verify the nonlinear consequence explicitly, put $D={\partial }_x$ and take the operator commutator in the order $[N,M]=NM-MN$. The product rule givesFor example, $[D^3,q]=3q_x{D}^2+3q_{xx}D+q_{xxx}$ and $[3qD,D^2]=-6q_x{D}^2-3q_{xx}D$, which explain the cancellations. Acting on $\phi$ yieldsThe integral equation has the large-real-$k$ normalization $\phi /E_k=1+O(k^{-1})$ under the stated contour regularity, so its eigenfunction is not zero for every $k$ at a fixed $(x,t)$. Consequently $q_t+q_{xxx}+6q{q}_x=0$.

Thus a time-independent contour and measure determine $\phi$ through a linear spectral integral equation; $q=-{\partial }_x{\int }_L\phi d\lambda$ then reconstructs the nonlinear solution. This is the asserted linearisation. It does not assert that sums of reconstructed nonlinear solutions are again solutions, and it depends on the given uniqueness and differentiability hypotheses.

Take the spectral measure to have $n$ point masses at $k_j=i{\kappa }_j$, with distinct ${\kappa }_j>0$ and positive weights $c_j$. SetEvaluating the linear integral equation at these points gives the explicit algebraic system and reconstructionThe sign is fixed by $i/(i{\kappa }_i+i{\kappa }_j)=1/({\kappa }_i+{\kappa }_j)$. No continuum integration remains.

There is also a useful determinant form. Define $A_{ij}=c_j{E}_i/({\kappa }_i+{\kappa }_j)$, $B=I+A$, $\tau =\det B$ and $\Lambda =\mathrm{diag}({\kappa }_1,\dots ,{\kappa }_n)$. Then $A_x=-\Lambda A$ and $\Lambda A+A\Lambda =E{c}^T$. Using the derivative of a determinant and cyclicity of the trace,Hence the finite-rank KdV dressing determinant givesPositivity ensures a regular real solution: the Cauchy matrix $C_{ij}=1/({\kappa }_i+{\kappa }_j)$ is the Gram matrix of the independent functions $e^{-{\kappa }_{i}s}$ on $s>0$, so it is positive definite. The matrix $A=\mathrm{diag}(E)C\mathrm{diag}(c)$ is diagonally similar to $\mathrm{diag}(\sqrt{cE})C\mathrm{diag}(\sqrt{cE})$. Its eigenvalues are positive, and $\tau >0$. The resulting reflectionless multisoliton solution has $n$ distinct scales and velocities; the weights fix their positions. Arbitrary signed or complex weights need not give a nonsingular real solution.