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Kortewegde Vries equation (KdV)
Kortewegde Vries equation  

Order  3rd 
Solvable  Exactly solvable 
The Cauchy problem  Solvable by inverse scattering transform 
Hierarchy  Kortewegde Vries hierarchy 
First introduced  Korteweg & de Vries, 1895 

Definition
$${u}_{t}+6u{u}_{x}+{u}_{xxx}=0$$
History
The equation is named for Diederik Korteweg and Gustav de Vries who studied it in 1895 (Korteweg & de Vries, 1895), though the equation first appears in work of Boussinesq (1877). The Kortewegde Vries equation was not studied much after that until Zabusky and Kruskal (1965) discovered numerically that its solutions seemed to decompose at large times into a collection of solitons.Solutions
Traveling wave
The Kortewegde Vries equation possesses a solution known from the end of the 19^{th} century. This solution can be expressed via Jacobi's elliptic functions. Let us look for solutions of the Kortewegde Vries equation$${u}_{t}+6u{u}_{x}+{u}_{xxx}=0$$
in the form of traveling wave$$u\left(x,t\right)=y\left(z\right),\text{\hspace{1em}}z=x{c}_{0}t.$$
Then the Kortewegde Vries equation will take the form$${y}_{zz}+3{y}^{2}{c}_{0}y+{c}_{1}=0.$$
$$\frac{{y}_{z}^{2}}{2}+{y}^{3}\frac{{c}_{0}{y}^{2}}{2}+{c}_{1}y+{c}_{2}=0.$$
This equation can be written in the form$${y}_{z}^{2}=2f\left(y\right),\text{\hspace{1em}}f\left(y\right)={y}^{3}\frac{{c}_{0}{y}^{2}}{2}+{c}_{1}y+{c}_{2}.$$
Function $f(y)$ can be written in the form$$f\left(y\right)=\left(y\alpha \right)\left(y\beta \right)\left(y\gamma \right)$$
where $\alpha $, $\beta $ and $\gamma $ are real roots of cubic equation$$f\left(y\right)={y}^{3}\frac{{c}_{0}{y}^{2}}{2}+{c}_{1}y+{c}_{2}=0.$$
One can express $\alpha $, $\beta $ and $\gamma $ via ${c}_{0}$, ${c}_{1}$ and ${c}_{2}$$$\alpha \beta \gamma ={c}_{2},$$
$$\alpha \beta +\alpha \gamma +\beta \gamma ={c}_{1},$$
$$\alpha +\beta +\gamma =\frac{{c}_{0}}{2}.$$
So we obtain$$\frac{dy}{\sqrt{2\left(\alpha y\right)\left(y\beta \right)\left(y\gamma \right)}}=dz.$$
Let us introduce new variables$$y=\alpha {p}^{2},\text{\hspace{1em}}p=\sqrt{\alpha \beta}q,\text{\hspace{1em}}{S}^{2}=\frac{\alpha \beta}{\alpha \gamma}.$$
Using these variables we get$$\underset{0}{\overset{q}{\int}}\frac{dx}{\sqrt{\left(1{x}^{2}\right)\left(1{S}^{2}{x}^{2}\right)}}=\sqrt{\frac{\alpha \gamma}{2}}\left(z{z}_{0}\right).$$
In the left hand side of the latter expression one can notice the elliptic integral of the first kind$$F\left(\mathrm{arcsin}q,S\right)=s{n}^{1}\left(q,S\right)=\sqrt{\frac{\alpha \gamma}{2}}\left(z{z}_{0}\right).$$
Taking this into account we have the following solution$$q=sn\left\{\sqrt{\frac{\alpha \gamma}{2}}\left(z{z}_{0}\right),{S}^{2}\right\},$$
where $\mathrm{sn}(z)$ is the Jacobi elliptic function. Returning from $q$ to $y$ we obtain$$y\left(z\right)=\alpha \left(\alpha \beta \right)s{n}^{2}\left\{\sqrt{\frac{\alpha \gamma}{2}}\left(z{z}_{0}\right),{S}^{2}\right\}.$$
Now let us take into account the wellknown relation between elliptic functions$$s{n}^{2}\left(z\right)+c{n}^{2}\left(z\right)=1$$
$$u\left(x,t\right)=\beta +\left(\alpha \beta \right)c{n}^{2}\left\{\sqrt{\frac{\alpha \gamma}{2}}\left(z{z}_{0}\right),{S}^{2}\right\}$$
with $0\le S\le 1$. This solution is a wave with speed ${c}_{0}$ and period $T$ defined by the following expressions$${c}_{0}=2\left(\alpha +\beta +\gamma \right),$$
$$T=2\sqrt{\frac{2}{\alpha \gamma}}{\displaystyle \underset{0}{\overset{1}{\int}}\frac{dx}{\sqrt{\left(1{x}^{2}\right)\left(1{S}^{2}{x}^{2}\right)}}=\sqrt{\frac{8}{\alpha \gamma}}K\left(S\right),}$$
where $K(S)$ is the complete elliptic integral of the first kind. Thus we obtained periodic wave described by the Kortewegde Vries equation. This wave sometimes is referred to as cnoidal wave for the look of the corresponding Jacobi elliptic function sign. In limiting case of small amplitude this solution turns into the wellknown sinusoidal wave.Soliton
If in the obtained periodic solution $\alpha >\beta =\gamma $ then ${S}^{2}=1$ and the period $T$ tends to infinity and we obtain solitary wave$$u\left(x,t\right)=y\left(z\right)=\beta +\left(\alpha \beta \right)c{h}^{2}\left\{\sqrt{\frac{\alpha \beta}{2}}\left(z{z}_{0}\right)\right\}.$$
Usually one sets $\beta =\gamma =0$, $\alpha =2{k}^{2}$ so the solution takes the form$$u\left(x,t\right)=2{k}^{2}c{h}^{2}\left\{k\left(x4{k}^{2}t\right)+{\chi}_{0}\right\},$$
where ${\chi}_{0}$ is an arbitrary constant. This solution describes the solitary wave observed by John Scott Russel in 1834 (Russel, 1845).Nsoliton solutions
Nsoliton solutions of the Kortewegde Vries equation can be obtained with the help of the Hirota direct method. See Application of Hirota method for the Kortewegde Vries equation.Selfsimilar solutions
Let us look for selfsimilar solutions of the Kortewegde Vries equation$${u}_{t}+6u{u}_{x}+{u}_{xxx}=0$$
in the form$$u\left(x,t\right)=\frac{1}{{\left(3t\right)}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{1ex}{$3$}\right.}}y\left(z\right),\text{\hspace{1em}}z=\frac{x}{{\left(3t\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{1ex}{$3$}\right.}}.$$
$${y}_{zzz}+6y{y}_{z}z{y}_{z}2y=0.$$
Now let us use the Miura transform$$y\left(z\right)={w}_{z}{w}^{2},\text{\hspace{1em}}w=w\left(z\right).$$
For new unknown function one obtains$${E}_{1}\left[w\right]={w}_{zzzz}2w{w}_{zzz}12w{w}_{z}^{2}6{w}^{2}{w}_{zz}+12{w}^{3}{w}_{z}$$
$$z{w}_{zz}+2zw{w}_{z}2{w}_{z}+2{w}^{2}=0.$$
The latter equation can be written in the form$${E}_{1}\left[w\right]=\left(\frac{d}{dz}2w\right)\frac{d}{dz}\left(\frac{{d}^{2}w}{d{z}^{2}}2{w}^{3}zw+\alpha \right)=0.$$
$${w}_{zz}2{w}^{3}zw+\alpha =0.$$
This is the second Painlevé equation. Solutions of this equation generally can be expressed via the socalled Painlevé transcendents. For integer $\alpha $ the second Painlevé equation possesses rational solutions; for halfinteger $\alpha $ its solutions can be expressed via Airy functions. Thus we've found out that selfsimilar solutions of the Kortewegde Vries equation can be expressed via solutions of the second Painlevé equation.Integrals of motion
Dicovery of solitons had put a question about integrals of motion for the Kortewegde Vries equation$${u}_{t}+6u{u}_{x}+{u}_{xxx}=0.$$
$$\frac{\partial u}{\partial t}+\frac{\partial}{\partial x}\left({u}_{xx}+3{u}^{2}\right)=0.$$
Therefore we obtain the first integral of motion of the Kortewegde Vries equation in the form$$\underset{\infty}{\overset{+\infty}{\int}}udx}=const.$$
In order to look for the other integrals of motion Gardner suggested to use the following Miura transform$$u\left(x,t\right)=w\epsilon {w}_{x}{\epsilon}^{2}{w}^{2},\text{\hspace{1em}}w=w\left(x,t\right).$$
One obtains the following correlation$${u}_{t}+6u{u}_{x}+{u}_{xxx}=\left(12{\epsilon}^{2}w\epsilon \frac{\partial}{\partial x}\right)\left[{w}_{t}+6\left(w{\epsilon}^{2}{w}^{2}\right)w{}_{x}+{w}_{xxx}\right].$$
One can notice that $u(x,t)$ doesn't contain parameter $\epsilon .$ It is possible if $w=w(x,t,\epsilon )$. Let us look for $w(x,t,\epsilon )$ in the form$$w\left(x,t,\epsilon \right)={\displaystyle \sum _{n=0}^{\infty}{w}_{n}\left(x,t\right){\epsilon}^{n}}.$$
From the equation$${w}_{t}+6\left(w{\epsilon}^{2}{w}^{2}\right){w}_{x}+{w}_{xxx}=0$$
we get$$\underset{\infty}{\overset{+\infty}{\int}}w\left(x,t,\epsilon \right)dx}=const.$$
Therefore we get the following set of relations$$\underset{\infty}{\overset{+\infty}{\int}}{w}_{n}\left(x,t\right)dx}=const,\text{\hspace{1em}}n=\mathrm{1,2,}\dots $$
$${w}_{0}=u,$$
$${w}_{1}={w}_{\mathrm{0,}x}={u}_{x},$$
$${w}_{2}={w}_{\mathrm{1,}x}+{w}_{0}^{2}={u}_{xx}+{u}^{2},$$
$${w}_{3}={w}_{\mathrm{2,}x}+2{w}_{0}{w}_{1}={u}_{xxx}+4u{u}_{x},$$
$${w}_{4}={w}_{\mathrm{3,}x}+2{w}_{0}{w}_{2}+{w}_{1}^{2}={u}_{xxxx}+6u{u}_{xx}+5{u}_{x}^{2}+2{u}^{3}.$$
All these expressions give integrals of motion for the Kortewegde Vries equation. One can continue the search procedure and obtain an infinite number of integrals of motion. This important property is typical for equations solvable by the inverse scattering transform.Miura transform and Lax pair
Miura found out that transform (now known as Miura transform)$$u={v}_{x}{v}^{2}$$
lets one express solutions of the Kortewegde Vries equation$${u}_{t}+6u{u}_{x}+{u}_{xxx}=0$$
via known solutions of the modified Kortewegde Vries equation$${v}_{t}6{v}^{2}{v}_{x}+{v}_{xxx}=0.$$
$${v}_{x}=u+\lambda +{v}^{2}.$$
Now, one can rewrite the modified Kortewegde Vries equation in the following way$$\begin{array}{l}{v}_{t}6{v}^{2}{v}_{x}+{v}_{xxx}={v}_{t}+\frac{\partial}{\partial x}\left[\left(\frac{\partial}{\partial x}+2v\right)\left({v}_{x}{v}^{2}\right)\right]=\\ ={v}_{t}+\frac{\partial}{\partial x}\left[\left(\frac{\partial}{\partial x}+2v\right)u\right].\end{array}$$
So we can treat Miura transform and the modified Kortewegde Vries equation as a system of equations$${v}_{x}=u+\lambda +{v}^{2}$$
$${v}_{t}=\frac{\partial}{\partial x}\left[\left(\frac{\partial}{\partial x}+2v\right)\left(u2\lambda \right)\right]$$
with compatibility condition$${\left({v}_{x}\right)}_{t}={\left({v}_{t}\right)}_{x}.$$
Let us introduce new unknown function $\psi (x,t)$$$v=\frac{{\psi}_{x}}{\psi}.$$
Then we get the following system of linear partial differential equations$${\psi}_{xx}+\left(u+\lambda \right)\psi =0$$
$${\psi}_{t}=\left(c\left(t\right)+{u}_{x}\right)\psi 2\left(u2\lambda \right){\psi}_{x}.$$
Obtained system can be written in the form$$\widehat{L}\psi =\lambda \psi $$
$${\psi}_{t}=\widehat{A}\psi ,$$
where $\widehat{L}$ is the Schrödinger operator$$\widehat{L}=\frac{{\partial}^{2}}{\partial {x}^{2}}u$$
and $\widehat{A}$ is an auxiliary evolutionary operator given by$$\widehat{A}=2\left(2\lambda u\right)\frac{\partial}{\partial x}+\left(c\left(t\right)+{u}_{x}\right).$$
Using compatibility condition$${\left({\psi}_{xx}\right)}_{t}={\left({\psi}_{t}\right)}_{xx}$$
$${\widehat{L}}_{t}+\left[\widehat{L},\widehat{A}\right]=0,$$
where $[\widehat{L},\widehat{A}]$ is the commutator defined as$$\left[\widehat{L},\widehat{A}\right]=\widehat{L}\widehat{A}\widehat{A}\widehat{L}.$$
The Kortewegde Vries equation is a nonlinear partial differential equation that can not be reduced to one linear equation. However it is equivalent to the system of linear equations. Using this system one can construct solution of the Cauchy problem for the Kortewegde Vries equation using the inverse scattering transform.The Cauchy problem
The Cauchy problem for the Kortewegde Vries equation can be solved using the inverse scattering transform (IST). This method was first discovered by Gardner, Greene, Kruskal and Miura in 1967 (Gardner, et al., 1967). See Application of the inverse scattering transform for the Kortewegde Vries equation.Lagrangian
The Kortewegde Vries equation$${u}_{t}+6u{u}_{x}+{u}_{xxx}=0$$
can be written in form$$\frac{\partial L}{\partial u}\frac{\partial}{\partial x}\frac{\partial L}{\partial {u}_{x}}\frac{\partial}{\partial t}\frac{\partial L}{\partial {u}_{t}}=0$$
where $L$ is a Lagrangian$$L=\frac{1}{2}{\psi}_{x}{\psi}_{t}+{\psi}_{x}^{3}\frac{1}{2}{\psi}_{xx}^{2}$$
with $u$ defined by$$u={\psi}_{x}.$$
Hamiltonian
The Kortewegde Vries equation$${u}_{t}+6u{u}_{x}+{u}_{xxx}=0$$
can be written in form$${u}_{t}=\frac{\partial}{\partial x}\frac{\delta H}{\delta u},$$
where $\frac{\delta}{\delta u}$ is a variational derivative given by$$\underset{\epsilon \to 0}{\mathrm{lim}}\frac{1}{\epsilon}\left(H\left[u+\epsilon \delta u\right]H\left[u\right]\right)={\displaystyle \underset{\infty}{\overset{+\infty}{\int}}\frac{\delta H}{\delta u}\delta udx}$$
and $H$ is a Hamiltonian function$$H={\displaystyle \underset{\infty}{\overset{\infty}{\int}}\left(\frac{1}{2}{u}_{x}^{2}{u}^{3}\right)}dx.$$
Thus we see that the Kortewegde Vries equation can be presented as an infinite dimensional Hamiltonian system.Applications and connections
The Kortewegde Vries equation is used as a model in fields as wide as: acoustic waves on a crystal lattice
 ionacoustic waves in a plasma
 long internal waves in a densitystratified ocean
 shallowwater waves with weakly nonlinear restoring forces
Variations
Modified Kortewegde Vries equation$${u}_{t}6{u}^{2}{u}_{x}+{u}_{xxx}=0$$
Generalized Kortewegde Vries equation$${u}_{t}+f\left(u\right){u}_{x}+{u}_{xxx}=0$$
Cylindrical Kortewegde Vries equation$${u}_{t}6u{u}_{x}+{u}_{xxx}+\frac{u}{2t}=0$$
See also
Kortewegde Vries hierarchy Modified Kortewegde Vries equationReferences
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External links
 Kortewegde Vries equation in Encyclopaedia of Mathematics
 Kortewegde Vries equation in EqWorld, the world of mathematical equations
 Kortewegde Vries equation in Wolfram MathWorld
 Kortewegde Vries equation in Scholarpedia
 Kortewegde Vries equation in Wikipedia, the free encyclopedia
 The Kortewegde Vries equation: history, exact solutions, and graphical representation by Klaus Brauer