Nonlinear partial differential equation

A nonlinear partial differential equation depends nonlinearly on the unknown function or one of its derivatives.

The nonlinear Schrödinger equation adds an amplitude-dependent interaction to Schrödinger-type wave evolution. Its Laplacian term disperses the complex wave amplitude, while the sign and form of the nonlinear term determine focusing or defocusing behavior. A one-dimensional cubic specialization is the focusing nonlinear Schrodinger equation; members of this broader family need not be integrable systems.

The complex Ginzburg–Landau equation is a nonlinear complex-amplitude evolution equation with linear and cubic terms and generally complex coefficients. The driven-condensate form ${\Psi }_t=(\alpha -\beta |\Psi {|}^2)\Psi +i({\nabla }^2-g|\Psi {|}^2+s)\Psi$ has gain/loss coefficients $\alpha ,\beta$ and conservative interaction/frequency coefficients $g,s$. A uniform nonzero state has number density $\alpha /\beta$ when $\alpha ,\beta >0$.

A complex Ginzburg–Landau equation describes a complex number disturbance amplitude through linear amplification, advection, complex diffusion equation terms, and often cubic amplitude saturation. A constant-coefficient example is ${\psi }_t+U{\psi }_x=\mu \psi +(1+i{c}_d){\psi }_{xx}-|\psi {|}^2\psi$. Here real $U,\mu ,c_d$ describe transport, linear growth rate and dispersive diffusion equation terms. Omitting the cubic term gives the linear complex Ginzburg-Landau equation.

For $A_t=A+(1+ib)A_{xx}-(1+ic)|A{|}^{2}A$, the uniform oscillation $A=e^{-ict}$ has phase diffusion coefficient $D=1+bc$. Its amplitude-phase dispersion relation is ${\lambda }^2+(2+2k^2)\lambda +2D{k}^2+(1+b^2)k^4=0$. Thus $D<0$ gives a long-wavelength modulational instability, whereas $D>0$ gives linear stability of all nonzero Fourier modes. On a fixed periodic domain, instability requires an allowed wavenumber with $k^2<-2D/(1+b^2)$.

Write $A=e^{-ict}(1+r)e^{i\varphi }$. The amplitude relaxes at rate two, while the phase is neutral under constant shifts. Eliminating the damped amplitude on long scales gives ${\varphi }_t=(1+bc){\varphi }_{xx}+(c-b)({\varphi }_x{)}^2-E{\varphi }_{xxxx}+\cdots$, where the linear phase branch has $E=b^2(1+c^2)/2$. Near $1+bc=0$, $E=(1+b^2)/2$ at leading order. Constant phase symmetry forbids undifferentiated phase terms; spatial parity of the leading amplitude equation forbids odd linear derivatives. Balancing negative phase diffusion against the fourth derivative produces the Kuramoto-Sivashinsky equation.

For real $\mu$ and $\alpha >0$, this complex amplitude equation has a destabilizing cubic term and a saturating quintic term. Nonzero uniform intensities solve $\mu +\alpha q-q^2=0$. The upper branch is radially stable and the lower branch radially unstable; constant phase is neutral because multiplication by $e^{i{\varphi }_0}$ preserves the equation. Stable conduction and the stable upper pattern coexist for $-{\alpha }^2/4<\mu <0$. At the fold, a zero radial eigenvalue conceals one-sided nonlinear instability, so strict stability cannot be inferred from eigenvalues alone.

A nontrivial stationary front of the real-coefficient cubic-quintic amplitude equation joining a nonzero pattern to zero must have constant phase. Writing $A=R{e}^{i\varphi }$ gives $R^2{\varphi }^{\prime }=J$; the finite first integral contains $J^2/(2R^2)$, so approaching zero forces $J=0$. Equal endpoint potentials then give $R_{-}^2=3\alpha /4$ and the unique parameter ${\mu }_M=-3{\alpha }^2/16$. The explicit profile is $A=e^{i{\varphi }_0}\sqrt{(3\alpha /4)/(1+e^{\sqrt{3}\alpha (x-x_0)/2})}$. This proves existence as well as the necessary equal-potential condition. The zero solution is excluded from the term front.

This nondispersive cubic complex Ginzburg–Landau equation balances linear growth against nonlinear damping. Its real-wavenumber Ginzburg-Landau plane wave has amplitude $\sqrt{\mu -k^2}$ when $\mu >k^2$. Relaxation within that single-mode amplitude family does not by itself establish stability against general perturbations.

With real $k,U$ and $\mu >k^2$, substitution into the nonlinear Ginzburg-Landau equation cancels the material derivative and gives $Q^2=\mu -k^2$. A complex $k$ makes the modulus spatially varying, so it generally does not admit this constant-amplitude nonlinear Ginzburg-Landau equation ansatz.

For $a>0$, the positive amplitude ordinary differential equation $R^{\prime }=aR-R^3$ has solution $R(t)=\sqrt{a}[1+(a/R_0^2-1)e^{-2at}{]}^{-1/2}$. This follows from the logistic differential equation for $R^2$. Every $R_0>0$ approaches $\sqrt{a}$ as a monotone function, and the positive distance to this limit decreases; an oscillating complex wave carrying this amplitude has no ordinary monotone ordering.

A normal mode $e^{i(kx-\omega t)}$ has dispersion relation $\omega =Uk+(c_d-i)k^2+i\mu$. The absolute wavenumber is the saddle where $d\omega /dk=0$. The Green function of the linear complex Ginzburg-Landau equation distinguishes convective hydrodynamic instability from absolute hydrodynamic instability.

On $x>0$ with a homogeneous wall condition, positive diffusion $\gamma$ and growth $\mu (x)={\mu }_0-ϵ\lambda x$, remove constant drift by $A=e^{Ux/(2\gamma )-i\omega t}f$. The scale $\xi =(ϵ\lambda /\gamma {)}^{1/3}x$ reduces the eigenvalue problem to the Airy ordinary differential equation. Decay selects $f=\mathrm{Ai}(\xi +z_n)$, where $\mathrm{Ai}(z_n)=0$ enforces the wall. The least negative zero $z_1$ sets the global instability threshold ${\mu }_0>U^2/(4\gamma )+|z_1|({ϵ}^2\gamma {\lambda }^2{)}^{1/3}$. A locally absolutely unstable pocket can therefore exist without an unstable global mode.

For $t>0$ this Green function is the response to a point impulse on the infinite line, using the continuous square root of $1+i{c}_d$. The Fourier transform multiplies the initial transform by $e^{\mu t-ikUt-(1+i{c}_d)k^{2}t}$; its Gaussian inverse gives the displayed expression. At fixed $x$, its exponential rate is $\mu -U^2/[4(1+c_d^2)]$, whereas along $x=Ut$ it is $\mu$. Thus a temporally growing flow with $0<\mu <U^2/[4(1+c_d^2)]$ is convectively rather than absolutely unstable.

For $[{\nabla }^2+\xi (1-|\psi {|}^2)]\psi =0$ and a regular unit-charge core $\psi =f(r)e^{i[\theta +\chi (r)]}$, write current velocity $u_r=2{\chi }^{\prime }$ when the time-dependent kinetic operator is $-{\nabla }^2$. Separating the imaginary part givesWith $f=ar+O(r^3)$ and no singular core flux, integration gives $u_r=-\mathrm{Im}\xi r/2+O(r^3)$. If $\xi =g{n}_{\infty }-i\alpha$, the slope is $\alpha /2$. The phase-gradient convention $k={\chi }^{\prime }$ has half this slope, $\alpha /4$. These are local regularity results, not a global vortex-existence assertion.