Integrable systems

Integrable systems admit unusually rich exact structure such as Lax pairs, conserved quantities, and symmetry reductions.

For a field evolving by a Hamiltonian partial differential equation, infinite-dimensional integrability combines sufficiently rich commuting conservation laws with a mechanism such as the inverse scattering transform or a spectral coordinate system that linearizes evolution. Lax equations supply spectral invariants; Lenard-Magri recursions supply commuting Hamiltonians when their solvability conditions hold. Infinite many conserved quantities alone do not prove completeness or an infinite-dimensional action-angle theorem: the function space, boundary conditions and analytic independence all matter.

A parameter-dependent system of differential relations maps a solution of one differential equation to a solution of another, or to a new solution of the same equation. Compatibility of the relations supplies the equations of motion. Integration constants select a particular transformed field. The Sine-Gordon Bäcklund transformation is an example of an auto-transformation and also generates conserved currents.

For a finite-dimensional Hamiltonian system, integrability is normally expressed by enough independent first integrals in mutual Poisson bracket involution. In classical field theories, an infinite hierarchy of compatible conserved quantities is the corresponding structure. For Sine-Gordon theory, Bäcklund transformations generate local conservation laws and explicit soliton solutions. Conservation of arbitrary quantities alone should not be confused with proof of their Hamiltonian involution.

An exact-solution method that transforms a nonlinear partial differential equation into equations bilinear in auxiliary Hirota tau functions. Finite exponential expansions of the Hirota tau functions produce soliton families. The auxiliary exponential coefficients need not all be positive.

An auxiliary function used in Hirota's bilinear method to encode a solution of an integrable partial differential equation. Products and ratios of tau functions replace nonlinear field variables; their differential equations become bilinear. In a Sine-Gordon multisoliton tau representation, two real tau functions encode the field as a continuous $4\arg (f+ig)$.

The periodic Hamiltonian chain $H=\frac{1}{2}\sum p_i^2+\sum e^{q_i-q_{i+1}}$, with cyclic indices, is a Liouville integrable system. Exponentials of neighbouring coordinate differences lead to a Lax pair and conserved characteristic coefficients.

The sinh-Gordon equation is an integrable nonlinear wave equation. Under the one-dimensional reduction $u=u(x)$ with $z=x+iy$, it becomesWriting $\varphi =2u$ gives ${\varphi }^{\prime \prime }=4\sinh \varphi$ and the first integral

In light-cone coordinates, the sine-Gordon equation can be written $u_{XT}=\sin u$. Its inverse scattering transform, solitons, breathers, and scaling symmetries make it an integrable system.

The transformationsform a one-parameter Lie point symmetry of $u_{XT}=\sin u$. The product $z=XT$ is a similarity variable, and a group-invariant solution $u=F(z)$ satisfies

For every $l>0$,solves $u_{XT}=\sin u$. The scaling $(X,T)\mapsto (e^{s}X,e^{-s}T)$ transforms $l$ by multiplication with $e^{-s}$.

A sine-Gordon breather is a spatially localized solution that is periodic in time and can be interpreted through a complex-conjugate pair of discrete scattering eigenvalues.

This is a breather of the Sine-Gordon equation; the general breather concept also occurs in other nonlinear systems.

The third Painlevé equation is the nonlinear second-order ordinary differential equationThe substitution $w=e^{iF}$ in the scaling reduction of the sine-Gordon equation gives the parameter choice $(\alpha ,\beta ,\gamma ,\delta )=(1/2,-1/2,0,0)$.

The focusing cubic nonlinear Schrödinger equation has the normalizationthe positive cubic term focuses the field and permits bright solitons on zero background.

For every $\kappa >0$ and $x_0\in \mathbb{R}$,solves $i{\psi }_t+{\psi }_{xx}+2|\psi {|}^2\psi =0$ and decays rapidly as $|x|\to \infty$.

The focusing and defocusing equationshave Hamiltoniansunder the convention $i{\psi }_t=\delta H_{\pm }/\delta \overline{\psi }$.

Writing a defocusing nonlinear Schrödinger field as $\psi =\sqrt{\rho }{e}^{iS}$ turns it into a continuity equation for $\rho$ and a velocity equation for $v=S_x$, including a quantum-pressure term.

In a common normalization, the defocusing case is $i{\psi }_t+{\psi }_{xx}-2|\psi {|}^2\psi =0$. The sign distinguishes it from the focusing nonlinear Schrödinger equation.

For the Robin boundary condition $q_x(0,t)=cq(0,t)$ with real $c$, the time Lax pair has a diagonal-conjugation symmetry under $k\mapsto -k$. It implies $A(-k)=A(k)$ and $B(-k)=-h(k)B(k)$. Together with the half-line NLS global relation, this eliminates unknown boundary scattering from an equivalent Riemann-Hilbert problem determined by the initial data and $c$.

Here $\Gamma =B^{♯}/(ad)$ and $\Delta =\mathcal{D}/(2k+ic)$. The difference follows algebraically from the half-line NLS global relation and the Robin symmetry. Multiplication by the spatial-temporal jump factor gives $e^{2ikx-4i{k}^2(T-t)}$, decaying in the second quadrant for $t<T$. A triangular analytic change of the Riemann-Hilbert problem removes this finite-horizon defect while retaining the reconstruction and transferring all genuine pole information to the known determinant. No unproved large-time decay of the boundary values is needed.

The effective boundary coefficient is $\tilde{\Gamma }=-(2k-ic)b^{♯}(-k)/(a\mathcal{D})$. All its ingredients come from initial scattering and the Robin boundary condition. Genuine zeros produce pole data; common zeros in numerator and denominator can instead be removable. It is incorrect to assert absence of boundary poles for every real Robin parameter.

Initial scattering gives $s$ and boundary-time scattering gives $S$. The defocusing reduction and unit determinants give the displayed matrix forms. The initial functions $a,b$ are analytic in the upper half-plane, while the boundary functions $A,B$ are bounded in the first and third quadrants. Their finite-horizon compatibility is the half-line NLS global relation.

Evaluating the Lax pair connection formula at $(0,T)$ gives $S^{-1}s=e^{2i{k}^{2}T{\hat{\sigma }}_3}{\mu }_3(0,T,k)$. The upper off-diagonal entry is the displayed relation, where $c_T=[{\mu }_3(0,T){]}_{12}$ is analytic in the upper half-plane and is $O(1/k)$. A finite-time relation must retain this final-state term.

For $\psi =e^{-iEt}f(x)$ with real $f$, the defocusing equation $i{\psi }_t+{\psi }_{xx}-2|\psi {|}^2\psi =0$ has first integralA nonzero smooth function tending to zero at both infinities cannot satisfy this identity: a nonzero extremum requires $f^2=E>0$, but then the right side is negative for sufficiently small nonzero $f$.

For the normalizationa dark soliton of speed $0\le U<1/\sqrt{2}$ is

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.

The inverse scattering transform evolves the scattering data of an auxiliary linear operator and reconstructs the potential from the evolved data.

Scattering data describe connections between normalized solutions of an auxiliary spectral equation, together with any discrete spectral points and their norming data. In an inverse scattering transform, the nonlinear potential is reconstructed from these data; boundary spectral connection matrices must additionally obey a global relation.

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.

Choose distinct ${\kappa }_j>0$ and weights $c_j>0$ at spectral points $i{\kappa }_j$. With $E_j=e^{-{\kappa }_{j}x+{\kappa }_j^{3}t}$ and $A_{ij}=c_j{E}_i/({\kappa }_i+{\kappa }_j)$, solve $(I+A)\Phi =E$. The reconstructed multisoliton solution is $q=-{\partial }_x(c^T\Phi )$. Since $\Lambda A+A\Lambda =E{c}^T$, cyclicity of the trace gives $c^T\Phi =-2{\partial }_x\log \det (I+A)$. The Cauchy matrix $1/({\kappa }_i+{\kappa }_j)$ is positive definite, because it is the Gram matrix of $e^{-{\kappa }_{i}s}$ on $s>0$; diagonal similarity then proves $\det (I+A)>0$. For one spectral point this yields $q=({\kappa }^2/2){\mathrm{sech}}^2[\kappa (x-{\kappa }^{2}t-x_0)/2]$.

Discrete scattering data consist of the discrete eigenvalues and norming constants associated with bound states of the auxiliary spectral problem. Their simple time evolution reconstructs solitons and breathers through the inverse problem.

For one-dimensional inverse scattering, the Gelfand-Levitan-Marchenko equationrecovers the transformation kernel $K$ from scattering data encoded by $F$. In the convention used for the Korteweg-De Vries equation above, the potential is

If $F(s)=c{e}^{-\chi s}$ with $\chi >0$, the Gelfand-Levitan-Marchenko equation has a separable kernel. Writing $K(x,y)=-B(x)e^{-\chi y}$ reduces the integral equation to one algebraic equation for $B(x)$ and reconstructs a single soliton.

Hamiltonians that split into terms $F_i(p_i,q_i)$ have $\{F_i,F_j\}=0$ for $i\ne j$, and their differentials are generically independent because they use disjoint coordinate pairs.

A Hamiltonian that is a sum of one-coordinate quadratic potentials plus linear terms becomes a collection of independent harmonic oscillators after translating each equilibrium position.

For $F=(p^2+W^2{q}^2+aq)/2$, shifting $Q=q+a/(2W^2)$ gives oscillator energy $E=F+a^2/(8W^2)$ and action $I=E/|W|$.

The action $(2\pi {)}^{-1}\oint pdq$ is the area enclosed by a periodic phase-plane orbit divided by $2\pi$.

Commuting mixed derivatives impose an equation on the coefficient matrices of an overdetermined auxiliary system.

An integrable partial differential equation has enough exact structure, such as a Lax pair, infinitely many conserved quantities, or an inverse scattering transform, to reduce its evolution to auxiliary linear or finite-dimensional data.

The Harry Dym equation is the integrable partial differential equation $p_t=p^3{p}_{xxx}$. It admits a Lax pair based on the second-order operator $L=-{\partial }_x{p}^2{\partial }_x$.

The Boussinesq equation is a nonlinear dispersive wave equation. One integrable normalization is

A Lax pair encodes a nonlinear equation as compatibility of two linear equations depending on an auxiliary spectral parameter.

For the modified Korteweg-De Vries equation with linear part $q_t-q_{xxx}$, the Volterra integral equation kernel is $e^{i[k(x-x^{\prime })-4k^3(t-t^{\prime })]{\hat{\sigma }}_3}$. In column one its off-diagonal modulus is $e^{2\mathrm{Im}k(x-x^{\prime })-8\mathrm{Im}{k}^3(t-t^{\prime })}$; column two has the reciprocal modulus. Define $D_1$ by signs $(+,+)$, $D_2$ by $(+,-)$, $D_3$ by $(-,+)$ and $D_4$ by $(-,-)$. The normalization points $(0,T)$, $(0,0)$ and $(\infty ,t)$ give column domains $(D_4,D_1)$, $(D_3,D_2)$ and $({\mathbb{C}}_{+},{\mathbb{C}}_{-})$, respectively. Finite-path eigenfunctions can be entire at fixed coordinates while failing to be bounded outside these spectral sectors.

Let $A$ have diagonal entries $\pm ({\lambda }^2+w+z/2)$ and off-diagonal entries $u\lambda +u^{\prime }$ and $-2w\lambda /u+2u^{\prime }w/u^2$. Let $B$ have diagonal entries $\pm \lambda /2$ and off-diagonal entries $u/2,-w/u$. Substitution into the isomonodromic deformation compatibility identity gives the displayed equations on domains with $u\ne 0$. Thus $(w/u^2{)}^{\prime }=0$, so $w=C{u}^2$ and $u^{\prime \prime }+C{u}^3+zu/2=0$, a rescaled zero-parameter Painlevé II equation when $C\ne 0$. The leading coefficient $-2w/u$ is required by cancellation of the quadratic spectral terms.

For the normalization points $(0,T)$, $(0,0)$ and spatial infinity, the Volterra integral equation kernel is $e^{-i[k(x-x^{\prime })+2k^2(t-t^{\prime })]{\hat{\sigma }}_3}$. Its lower entry in column one has factor $e^{2i[k(x-x^{\prime })+2k^2(t-t^{\prime })]}$; column two has the opposite factor. With quadrants numbered counterclockwise, the column domains are $(D_2,D_3)$, $(D_1,D_4)$ and $({\mathbb{C}}_{-},{\mathbb{C}}_{+})$, respectively.

For $D_i={\partial }_i+A_i$ and Higgs multiplication $\Phi$, setWrite $a=F_{12}-D_3\Phi$, $b=F_{31}-D_2\Phi$, $c=F_{23}-D_1\Phi$, where $D_i\Phi ={\partial }_i\Phi +[A_i,\Phi ]$. Direct expansion gives $[L_0,L_1]=-b+ic-2i\lambda a-{\lambda }^2(b+ic)$. Its vanishing for every complex $\lambda$ is therefore equivalent to $a=b=c=0$, the Bogomolny equations. It is a parameter-dependent zero-curvature representation.

For symmetric $L$ and antisymmetric $A$ satisfying $\dot{L}=LA-AL$, solve $\dot{Q}=-AQ$, $Q(0)=I$. Then $Q$ is orthogonal and $Q^{T}LQ$ is constant. Hence every eigenvalue and every coefficient of the characteristic polynomial of $L$ is conserved.

For periodic $U(x,t,\lambda )$ with period $L$, solve $W_x=UW$, $W(0)=I$. The monodromy matrix is $w=W(L)$. If $U,V$ satisfy the zero-curvature condition and are periodic, then $W_t=VW-WV(0)$, hence $w_t=[V(0),w]$. The matrix trace of each power of $w$ is a first integral.

A spectral parameter is an auxiliary complex parameter in a linear system whose compatibility encodes a nonlinear integrable equation. Its analytic dependence organizes conserved quantities and scattering data.

For covariant derivatives on Euclidean four-space, one anti-self-dual Yang-Mills Lax pair isThe condition $[L,M]=0$ for every $\zeta \in {\mathbb{CP}}^1$ is equivalent to anti-self-duality of the curvature.

For metric $2(dwd\bar{w}+dzd\bar{z})$, the commutator is $[L,M]=F_{wz}+\lambda (F_{w\bar{w}}+F_{z\bar{z}})+{\lambda }^2{F}_{\bar{w}\bar{z}}$. Vanishing for every spectral parameter $\lambda$ is exactly the Anti-self-dual Yang-Mills equations. The derivative parts span a totally null two-plane in the complexified tangent space. Thus an invertible fundamental solution of $L\Psi =M\Psi =0$ exists locally exactly when the three gauge curvature coefficients vanish; compatibility of one possibly zero vector alone is not enough.

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The Airy equation is the linear dispersive partial differential equation $q_t+q_{xxx}=0$. Under the spatial Fourier transform, each mode evolves by multiplication by $e^{i{k}^{3}t}$.

For positive time its fundamental solution is $(3t{)}^{-1/3}\mathrm{Ai}(x/(3t{)}^{1/3})$, connecting this evolution equation to the Airy function without identifying the two concepts.

Let $y$ solve the Airy ordinary differential equation $y^{\prime \prime }(s)=sy(s)$. For $t>0$, set $s=x/(3t{)}^{1/3}$ and define the displayed similarity solution. Differentiating the profile equation gives $y^{\prime \prime \prime }=y+s{y}^{\prime }$, so the chain rule yieldsTheir sum is zero, proving the Airy equation $u_t+u_{xxx}=0$. Every real linear combination of the two standard Airy functions gives such a solution. This is a solution of the linear dispersive equation; the nonlinear Korteweg-De Vries equation includes an extra term and does not follow from the same arbitrary profile construction.

For dispersion $w(k)=-i(k^3+k)$, the other roots of $w(\nu )=w(k)$ satisfy ${\nu }^2+k\nu +k^2+1=0$. In $D_{+}=\{\mathrm{Im}k>0,\mathrm{Re}w(k)<0\}$ both roots lie in the lower half-plane. Interpolating the unknown linear polynomial $G_2+i\nu G_1$ at them eliminates two boundary traces. The resulting symmetric expression has removable singularities where the roots coalesce.

For $Q_T+Q_{XXX}+a{Q}_{XX}+b{Q}_X+cQ=0$, an exponential gauge removes the second derivative. With ${\kappa }^2=a^2/3-b>0$ and $\sigma =-c+ab/3-2a^3/27$, the remaining equation is $q_t+q_{xxx}-q_x=0$. Negative $a$ improves spatial decay of the transformed initial data; positive $a$ can destroy the convergence of its Half-range Fourier transform.

For the Airy equation with opposite dispersive sign, $q_t-q_{xxx}=0$ on $x>0$, let $Q$ be the initial Half-range Fourier transform and $F_j(k,t)={\int }_0^t{e}^{i{k}^{3}s}{\partial }_x^{j}q(0,s)ds$. Three integrations by parts give the displayed global relation. The spatial transform is holomorphic for $\mathrm{Im}k<0$, whereas the finite-time spectral boundary transforms are entire functions. The symmetry $k\mapsto \alpha k$, ${\alpha }^3=1$, preserves their time exponent.

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For $q_t-q_{xxx}=0$ on $x>0$, evaluate the global relation at $k=-ir/2$, where $\mathrm{Re}p>0$ and the principal cube root $r=p^{1/3}$ has positive real part. The time exponent becomes $-p$, giving the displayed Laplace transform identity. Thus the second-derivative trace is determined once the value and first-derivative traces are prescribed. The homogeneous transformed equation $u^{\prime \prime \prime }-pu=0$ has exactly two spatially decaying roots, so two independent admissible conditions are required. One condition leaves a free decaying mode; three independent conditions generally violate the global constraint.

For $L=-d^2/d{x}^2+x$ on $(0,1)$, with homogeneous conditions $v(0)=0$, $v^{\prime }(1)=0$, integration by parts gives $⟨v,Lv⟩={\int }_0^1(v^{\prime 2}+x{v}^2)dx>0$ for nonzero $v$. The affine energy $J[u]=\frac{1}{2}{\int }_0^1(u^{\prime 2}+x{u}^2)dx$, minimized over $H^1$ functions with $u(0)=a$, yields $Lu=0$ and the natural boundary condition $u^{\prime }(1)=0$. The inequality $\Vert v{\Vert }_2^2\le \frac{1}{2}\Vert v^{\prime }{\Vert }_2^2$ on $v(0)=0$ proves that the form is a coercive bilinear form; Lax-Milgram theorem supplies a unique critical point, and $J[u+v]-J[u]=\frac{1}{2}\int (v^{\prime 2}+x{v}^2)$ proves that it is the unique minimum.

The scalar linear partial differential equation $u_t+u_x+u_{xxx}=0$ is an Airy equation with a first-order transport term. It is a linearization of the Korteweg-De Vries equation with an advective term retained. It is distinct from the viscous incompressible-flow Stokes equation. On the whole line its spatial Fourier transform evolves by $e^{i(k^3-k)t}$.

For a time Laplace transform parameter with $\mathrm{Re}p>0$, the spatial characteristic roots satisfy $r^3+r+p=0$. Exactly one root $r_{-}$ has negative real part and two have positive real part. Indeed an imaginary root would force $p=-i(k-k^3)$ to be imaginary. The count is therefore constant on the connected right half-plane; for real $p>0$, the cubic has one negative real root and a conjugate pair with positive real parts. Repeated roots would satisfy $3r^2+1=0$, which again puts $p$ on the imaginary axis. Thus $r_{-}(p)$ is a nonzero holomorphic function in the right half-plane. Only $e^{r_{-}x}$ decays at positive spatial infinity.

For $\mathrm{Re}p>0$, let $r_{-}$ and $r_1,r_2$ be the characteristic roots of $r^3+r+p=0$, split by the signs of their real parts. Put $d_j=3r_j^2+1$. The real-line Green function of $p+{\partial }_x+{\partial }_x^3$ isThe reciprocal-polynomial root identities imply continuity of $R_p,R_p^{\prime }$ and jump $R_p^{\prime \prime }(0+)-R_p^{\prime \prime }(0-)=1$. The distributional derivative therefore gives $(p+{\partial }_s+{\partial }_s^3)R_p={\delta }_0$. The transport term changes the roots: the pure Airy resolvent kernel with $r^3+p=0$ cannot be substituted unchanged.