Korteweg-De Vries equation

The Korteweg-De Vries equationis an integrable nonlinear dispersive equation. Its sign convention here admits negative solitary waves.

For the sign convention $u_t-6u{u}_x+u_{xxx}=0$ and any $\kappa >0$, the displayed negative solitary wave travels at speed $4{\kappa }^2$. To verify it, put $\xi =x-ct$ and write $u=U(\xi )$. The equation becomes $(-cU-3U^2+U^{\prime \prime }{)}^{\prime }=0$. A decaying profile has zero integration constant, so $U^{\prime \prime }=cU+3U^2$. Direct differentiation of the displayed profile gives $U^{\prime \prime }=4{\kappa }^{2}U+3U^2$, proving the result. Its depth is $2{\kappa }^2$, so narrower and deeper pulses travel faster. Changing the sign of the nonlinear term reverses the amplitude sign. This wave belongs to the soliton family of the Korteweg-De Vries equation.

One normalization of the first Hamiltonian structure uses the constant differential Poisson operator $J=-2{\partial }_x$. For functionals with variational derivatives $f,g$, its Poisson bracket is $\int fJgdx=-2\int f{g}^{\prime }dx$ under periodic or vanishing-boundary conventions. Thus the field bracket is $\{u(x),u(y)\}=-2{\partial }_x\delta (x-y)$. The factor and sign change with normalization; the constant first-order operator is the invariant structural feature.

The cylindrical KdV equation in this normalization is $q_t+q_{xxx}+q{q}_x+q/(3t)=0$, for $t\ne 0$. The additional time-dependent term distinguishes it from the ordinary KdV equation. A scaling-invariant solution has $q=t^{-2/3}F(x{t}^{-1/3})$. Substitution gives $F^{(3)}+F{F}^{\prime }-(\xi F{)}^{\prime }/3=0$. Integration reduces the similarity problem to $F^{\prime \prime }+F^2/2-\xi F/3=C$. The integration constant depends on supplementary conditions and cannot be set to zero without them. Use real cube roots separately on either sign of $t$.

This nonlinear dispersive equation replaces the quadratic KdV nonlinearity by a cubic one. In the displayed sign convention, the Miura transformation maps its solutions to $u_t-6u{u}_x+u_{xxx}=0$. Its scaling symmetry is $(x,t,v)\mapsto (e^{s}x,e^{3s}t,e^{-s}v)$.

For the three corner-normalized Lax pair eigenfunctions, let $s={\mu }_3(0,0,k)$ and $S=e^{4i{k}^{3}T{\hat{\sigma }}_3}{\mu }_2(0,T,k{)}^{-1}$. Their constant connection matrices give the displayed global relation. If their second columns are $(b,a{)}^T$ and $(B,A{)}^T$, its upper-right entry is $Ab-Ba=e^{8i{k}^{3}T}{\mu }_{3,12}(0,T,k)$. The right side is analytic in the lower half-plane. In the linear limit this is the backward-sign Airy half-line global relation at Fourier parameter $2k$, coupling all three boundary traces rather than allowing three independent boundary conditions.

Direct differentiation givesConsequently this transformation takes solutions of the modified Korteweg-De Vries equation to solutions of the Korteweg-De Vries equation in these sign conventions. Both summands in $u$ have the same scaling weight, so the modified-equation scaling induces $(x,t,u)\mapsto (e^{s}x,e^{3s}t,e^{-2s}u)$.

The inverse-scattering spectral problem associated with this KdV convention isFordifferentiation gives the Wronskian identity

A Jost solution of a one-dimensional Schrödinger 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.