WikiWaves - User contributions [en]

KdV Equation Derivation

2008-10-14T15:24:09Z

Mala058:

We consider the method of derivation of KdV Equation in the concept of [http://www.wikiwaves.org/index.php/Nonlinear_Shallow_Water_Waves Nonlinear Shallow Water Waves].

==Introduction==

In the analysis of [http://www.wikiwaves.org/index.php/Nonlinear_Shallow_Water_Waves Nonlinear Shallow Water Waves] equations we see that there are two important geometrical parameters, <math>\epsilon = \frac{h}{\lambda}</math> and <math>\alpha=\frac{a}{h}</math> involved. By choosing appropriate magnitudes for <math>\epsilon</math> and <math>\alpha</math>, we can consider a theory in which dispersion and nonlinearity are in balance. The Korteweg-de Vries Equation verifies the relation between dispersion and nonlinearity properties.

==Derivation==

We begin with the equations for waves on water,

<center>
<math>
\begin{matrix}
&\Phi_{xx} + \Phi_{yy} &= 0 \quad &-\infin<x<\infin, 0 \le y \le \eta(x,t) \\
\end{matrix}
</math>
</center>

Provided that at <math>y=\eta(x,t)=h+aH(x,t)</math> we have,

<center><math>
\begin{matrix}
&\Phi_{y} &= &\eta_t + \Phi_x \eta_x \\
&\Phi_t + \frac{1}{2}({\Phi_x}^2 + {\Phi_y}^2) + g\eta &= &B(t)\\
&\Phi_y = 0 &, &y = 0
\end{matrix}
</math></center>

To make these equations dimensionless, we use the scaled variables,
<center><math>
\bar{x}=\frac{x}{\lambda}, \quad \bar{y}=\frac{y}{h}, \quad \bar{\Phi}=\frac{h\Phi}{\lambda a \sqrt{gh}}, \quad \bar{t}=\frac{t\sqrt{gh}}{\lambda}
</math></center>

where <math>\sqrt{gh}</math> is defined as linear wave speed in shallow water. Hence the dimensionless system is,

<center><math>
\begin{matrix}
&\epsilon^2 {\bar{\Phi}}_{\bar{x}\bar{x}} + {\bar{\Phi}}_{\bar{y}\bar{y}} &= &0 \\ \\
&{\bar{\Phi}}_{\bar{y}} &= &\epsilon^2(H_{\bar{t}}+\alpha {\bar{\Phi}}_{\bar{x}} H_{\bar{x}}) \\ \\
&{\bar{\Phi}}_{\bar{t}} + \frac{1}{2}\alpha ({{\bar{\Phi}}_{\bar{x}}}^2 + \epsilon^2 {{\bar{\Phi}}_{\bar{y}}}^2) + H &= &(B(t)-gh) / ag \\ \\
&{\bar{\Phi}}_{\bar{y}} = 0 &, &\bar{y} = 0
\end{matrix}
</math></center>

where <math>\epsilon = \frac{h}{\lambda}</math> and <math>\alpha=\frac{a}{h}</math> are two small parameters which are given in this problem.

In the next step we use the transform <math>\bar{\Phi} \to \bar{\Phi} + \int\limits_{0}^{\bar{t}}(\frac{B(s) - gh}{ag})ds</math> and introduce further transformation to remove <math>\epsilon</math> from the equations,

<center><math>
z = \frac{\alpha^{1 / 2}}{\epsilon}(\bar{x}-\bar{t}), \quad \tau = \frac{\alpha^{3/2}}{\epsilon}\bar{t}, \quad \Psi = \frac{\alpha^{1/2}}{\epsilon}\bar{\Phi}
</math></center>

The key idea is that <math>\frac{\alpha^{1 / 2}}{\epsilon}</math> is <math>O(1)</math>.

Hence,

<center><math>
\begin{matrix}
&\alpha \Psi_{zz} + \Psi_{\bar{y}\bar{y}} = 0 & -\infin < z <\infin , 0 \le \bar{y} \le 1 + \alpha H(z,\tau) &(1) \\ \\
&\Psi_{\bar{y}} = \alpha (-H_z+\alpha H_{\tau} + \alpha \Psi_z H_z) & y=1+ \alpha H(z,\tau) &(2) \\ \\
&H - \Psi_z + \alpha \Psi_{\tau} + \frac{1}{2} ({\Psi_{\bar{y}}}^2+\alpha {\Psi_z}^2)=0 &y=1+ \alpha H(z,\tau) &(3) \\ \\
&\Psi_{\bar{y}} = 0 &\bar{y}=0 &(4)
\end{matrix}
</math></center>

The boundary condition (4) expresses <math>\Psi</math> at the flat bed, <math>\bar{y}=0</math>. The boundary condition (3) is Bernoulli equation and (2) is kinematic boundary condition. Now we use asymptotic expansions of the form,

<center><math>
\begin{matrix}
&\Psi &= &\Psi_0 + \alpha \Psi_1 + {\alpha}^2 \Psi_2 + o({\alpha}^2) &(5)\\ \\
&H &= &H_0 + \alpha H_1 + o(\alpha) &(6)
\end{matrix}
</math></center>

to derive an equation for each <math>H_i</math> according to the boundary conditions (2) to (4).

* Derivation of <math>H_i</math>'s:

Substituting (5) and (6), (1) must be true for all powers of <math>\alpha</math>. Therefore,

<center><math>
\begin{matrix}
&O(\alpha^0) &: &\Psi_{0, \bar{y}\bar{y}} = 0 &\rArr &\Psi_0 = B_0(z, \tau) \\ \\
&O(\alpha) &: &\Psi_{1, \bar{y}\bar{y}} = -\Psi_{0, zz} &\rArr &\Psi_1 = -\frac{1}{2}{\bar{y}}^2 B_{0, zz}+B_1(z, \tau) \\ \\
&O(\alpha^2) &: &\Psi_{2, \bar{y}\bar{y}} = -\Psi_{1, zz} &\rArr &\Psi_2 = \frac{1}{24}{\bar{y}}^4B_{0,zzzz}-\frac{1}{2}{\bar{y}}^2 B_{1,zz}+ B_2(z, \tau)
\end{matrix}
</math></center>

Now at leading order the Bernoulli and kinematic equations, (3) and (2), gives,

<center><math>
\begin{matrix}
&H_0(z,\tau) = \Psi_{0,z} = B_{0,z} &(a) \\ \\
&H_1-B_{1,z}+\frac{1}{2}B_{0,zzz}+B_{0,\tau}+\frac{1}{2}B^2_{0,z} = 0 &(b) \\ \\
&-H_0B_{0,zz}+\frac{1}{6}B_{0,zzzz}-B_{1,zz} = -H_{1,z}+H_{0,\tau}+B_{0,z}H_{0,z} &(c)
\end{matrix}
</math></center>

Differentiating (b) and eliminating <math>H_1</math> and <math>B_1</math> from (c) allow us to write,

<center><math>
-H_0B_{0,zz}-\frac{1}{3}B_{0,zzzz}-B_{0,z\tau}-B_{0,z}B_{0,zz} = H_{0,\tau}+B_{0,z}H_{0,z}
</math></center>

Finally, (a) gives <math>B_0</math> in terms of <math>H_0</math> and hence

<center><math>
2H_{0,\tau}+3H_0H_{0,z}+\frac{1}{3}H_{0,zzz}=0
</math></center>

which is named Korteweg-de Vries (KdV) equation.

==Interpretation==

KdV equation includes dispersive effects through the term <math>H_{0,zzz}</math> and nonlinear effects through the term <math>H_0H_{0,z}</math> and governs the behavior of the small amplitude waves, with <math>\alpha<<1</math>. It is reasonable to ask when and where the independent variables, <math>z</math> and <math>\tau</math>, are of <math>O(1)</math> in order to determine more precisely the region in physical space where the KdV equation is valid as an approximation of the actual flow. According to the definition of <math>z</math> and <math>\tau</math>, if <math>\alpha=O(\epsilon^2)</math>, then <math>\bar{t}>>1</math> and <math>\bar{x}=\bar{t}+O(1)</math>. This leads us to interpret any waveform that arises as a solution of the KdV equation as the large time limit of an initial value problem.

For solution of KdV equation please refer [http://www.wikiwaves.org/index.php/KdV_Equation_Solutions here.]

[[Category:789]]

KdV Equation Derivation

2008-10-14T15:22:42Z

Mala058:

We consider the method of derivation of KdV Equation in the concept of [http://www.wikiwaves.org/index.php/Nonlinear_Shallow_Water_Waves Nonlinear Shallow Water Waves].

== Introduction ==

In the analysis of [http://www.wikiwaves.org/index.php/Nonlinear_Shallow_Water_Waves Nonlinear Shallow Water Waves] equations we see that there are two important geometrical parameters, <math>\epsilon = \frac{h}{\lambda}</math> and <math>\alpha=\frac{a}{h}</math>, are involved. By choosing appropriate magnitudes for <math>\epsilon</math> and <math>\alpha</math>, we can consider a theory in which dispersion and nonlinearity are in balance. The Korteweg-de Vries Equation verifies the relation between dispersion and nonlinearity properties.

== Derivation ==

We begin with the equations for waves on water,

<center>
<math>
\begin{matrix}
&\Phi_{xx} + \Phi_{yy} &= 0 \quad &-\infin<x<\infin, 0 \le y \le \eta(x,t) \\
\end{matrix}
</math>
</center>

Provided that at <math>y=\eta(x,t)=h+aH(x,t)</math> we have,

<center><math>
\begin{matrix}
&\Phi_{y} &= &\eta_t + \Phi_x \eta_x \\
&\Phi_t + \frac{1}{2}({\Phi_x}^2 + {\Phi_y}^2) + g\eta &= &B(t)\\
&\Phi_y = 0 &, &y = 0
\end{matrix}
</math></center>

To make these equations dimensionless, we use the scaled variables,
<center><math>
\bar{x}=\frac{x}{\lambda}, \quad \bar{y}=\frac{y}{h}, \quad \bar{\Phi}=\frac{h\Phi}{\lambda a \sqrt{gh}}, \quad \bar{t}=\frac{t\sqrt{gh}}{\lambda}
</math></center>

where <math>\sqrt{gh}</math> is defined as linear wave speed in shallow water. Hence the dimensionless system is,

<center><math>
\begin{matrix}
&\epsilon^2 {\bar{\Phi}}_{\bar{x}\bar{x}} + {\bar{\Phi}}_{\bar{y}\bar{y}} &= &0 \\ \\
&{\bar{\Phi}}_{\bar{y}} &= &\epsilon^2(H_{\bar{t}}+\alpha {\bar{\Phi}}_{\bar{x}} H_{\bar{x}}) \\ \\
&{\bar{\Phi}}_{\bar{t}} + \frac{1}{2}\alpha ({{\bar{\Phi}}_{\bar{x}}}^2 + \epsilon^2 {{\bar{\Phi}}_{\bar{y}}}^2) + H &= &(B(t)-gh) / ag \\ \\
&{\bar{\Phi}}_{\bar{y}} = 0 &, &\bar{y} = 0
\end{matrix}
</math></center>

where <math>\epsilon = \frac{h}{\lambda}</math> and <math>\alpha=\frac{a}{h}</math> are two small parameters which are given in this problem.

In the next step we use the transform <math>\bar{\Phi} \to \bar{\Phi} + \int\limits_{0}^{\bar{t}}(\frac{B(s) - gh}{ag})ds</math> and introduce further transformation to remove <math>\epsilon</math> from the equations,

<center><math>
z = \frac{\alpha^{1 / 2}}{\epsilon}(\bar{x}-\bar{t}), \quad \tau = \frac{\alpha^{3/2}}{\epsilon}\bar{t}, \quad \Psi = \frac{\alpha^{1/2}}{\epsilon}\bar{\Phi}
</math></center>

The key idea is that <math>\frac{\alpha^{1 / 2}}{\epsilon}</math> is <math>O(1)</math>.

Hence,

<center><math>
\begin{matrix}
&\alpha \Psi_{zz} + \Psi_{\bar{y}\bar{y}} = 0 & -\infin < z <\infin , 0 \le \bar{y} \le 1 + \alpha H(z,\tau) &(1) \\ \\
&\Psi_{\bar{y}} = \alpha (-H_z+\alpha H_{\tau} + \alpha \Psi_z H_z) & y=1+ \alpha H(z,\tau) &(2) \\ \\
&H - \Psi_z + \alpha \Psi_{\tau} + \frac{1}{2} ({\Psi_{\bar{y}}}^2+\alpha {\Psi_z}^2)=0 &y=1+ \alpha H(z,\tau) &(3) \\ \\
&\Psi_{\bar{y}} = 0 &\bar{y}=0 &(4)
\end{matrix}
</math></center>

The boundary condition (4) expresses <math>\Psi</math> at the flat bed, <math>\bar{y}=0</math>. The boundary condition (3) is Bernoulli equation and (2) is kinematic boundary condition. Now we use asymptotic expansions of the form,

<center><math>
\begin{matrix}
&\Psi &= &\Psi_0 + \alpha \Psi_1 + {\alpha}^2 \Psi_2 + o({\alpha}^2) &(5)\\ \\
&H &= &H_0 + \alpha H_1 + o(\alpha) &(6)
\end{matrix}
</math></center>

to derive an equation for each <math>H_i</math> according to the boundary conditions (2) to (4).

* Derivation of <math>H_i</math>'s:

Substituting (5) and (6), (1) must be true for all powers of <math>\alpha</math>. Therefore,

<center><math>
\begin{matrix}
&O(\alpha^0) &: &\Psi_{0, \bar{y}\bar{y}} = 0 &\rArr &\Psi_0 = B_0(z, \tau) \\ \\
&O(\alpha) &: &\Psi_{1, \bar{y}\bar{y}} = -\Psi_{0, zz} &\rArr &\Psi_1 = -\frac{1}{2}{\bar{y}}^2 B_{0, zz}+B_1(z, \tau) \\ \\
&O(\alpha^2) &: &\Psi_{2, \bar{y}\bar{y}} = -\Psi_{1, zz} &\rArr &\Psi_2 = \frac{1}{24}{\bar{y}}^4B_{0,zzzz}-\frac{1}{2}{\bar{y}}^2 B_{1,zz}+ B_2(z, \tau)
\end{matrix}
</math></center>

Now at leading order the Bernoulli and kinematic equations, (3) and (2), gives,

<center><math>
\begin{matrix}
&H_0(z,\tau) = \Psi_{0,z} = B_{0,z} &(a) \\ \\
&H_1-B_{1,z}+\frac{1}{2}B_{0,zzz}+B_{0,\tau}+\frac{1}{2}B^2_{0,z} = 0 &(b) \\ \\
&-H_0B_{0,zz}+\frac{1}{6}B_{0,zzzz}-B_{1,zz} = -H_{1,z}+H_{0,\tau}+B_{0,z}H_{0,z} &(c)
\end{matrix}
</math></center>

Differentiating (b) and eliminating <math>H_1</math> and <math>B_1</math> from (c) allow us to write,

<center><math>
-H_0B_{0,zz}-\frac{1}{3}B_{0,zzzz}-B_{0,z\tau}-B_{0,z}B_{0,zz} = H_{0,\tau}+B_{0,z}H_{0,z}
</math></center>

Finally, (a) gives <math>B_0</math> in terms of <math>H_0</math> and hence

<center><math>
2H_{0,\tau}+3H_0H_{0,z}+\frac{1}{3}H_{0,zzz}=0
</math></center>

which is named Korteweg-de Vries (KdV) equation.

== Interpretation ==
KdV equation includes dispersive effects through the term <math>H_{0,zzz}</math> and nonlinear effects through the term <math>H_0H_{0,z}</math> and governs the behavior of the small amplitude waves, with <math>\alpha<<1</math>. It is reasonable to ask when and where the independent variables, <math>z</math> and <math>\tau</math>, are of <math>O(1)</math> in order to determine more precisely the region in physical space where the KdV equation is valid as an approximation of the actual flow. According to the definition of <math>z</math> and <math>\tau</math>, if <math>\alpha=O(\epsilon^2)</math>, then <math>\bar{t}>>1</math> and <math>\bar{x}=\bar{t}+O(1)</math>. This leads us to interpret any waveform that arises as a solution of the KdV equation as the large time limit of an initial value problem.

For solution of KdV equation please refer [http://www.wikiwaves.org/index.php/KdV_Equation_Solutions here.]

[[Category:789]]

KdV Equation Derivation

2008-10-14T14:56:58Z

Mala058:

We consider the method of derivation of KdV Equation in the concept of [http://www.wikiwaves.org/index.php/Nonlinear_Shallow_Water_Waves Nonlinear Shallow Water Waves].

== Introduction ==

In the analysis of [http://www.wikiwaves.org/index.php/Nonlinear_Shallow_Water_Waves Nonlinear Shallow Water Waves] equations we see that there are two important geometrical parameters, <math>\epsilon = \frac{h}{\lambda}</math> and <math>\alpha=\frac{a}{h}</math>, are involved. By choosing appropriate magnitudes for <math>\epsilon</math> and <math>\alpha</math>, we can consider a theory in which dispersion and nonlinearity are in balance. The Korteweg-de Vries Equation verifies the relation between dispersion and nonlinearity properties.

== Derivation ==

We begin with the equations for waves on water,

<center>
<math>
\begin{matrix}
&\Phi_{xx} + \Phi_{yy} &= 0 \quad &-\infin<x<\infin, 0 \le y \le \eta(x,t) \\
\end{matrix}
</math>
</center>

Provided that at <math>y=\eta(x,t)=h+aH(x,t)</math> we have,

<center><math>
\begin{matrix}
&\Phi_{y} &= &\eta_t + \Phi_x \eta_x \\
&\Phi_t + \frac{1}{2}({\Phi_x}^2 + {\Phi_y}^2) + g\eta &= &B(t)\\
&\Phi_y = 0 &, &y = 0
\end{matrix}
</math></center>

To make these equations dimensionless, we use the scaled variables,
<center><math>
\bar{x}=\frac{x}{\lambda}, \quad \bar{y}=\frac{y}{h}, \quad \bar{\Phi}=\frac{h\Phi}{\lambda a \sqrt{gh}}, \quad \bar{t}=\frac{t\sqrt{gh}}{\lambda}
</math></center>

where <math>\sqrt{gh}</math> is defined as linear wave speed in shallow water. Hence the dimensionless system is,

<center><math>
\begin{matrix}
&\epsilon^2 {\bar{\Phi}}_{\bar{x}\bar{x}} + {\bar{\Phi}}_{\bar{y}\bar{y}} &= &0 \\ \\
&{\bar{\Phi}}_{\bar{y}} &= &\epsilon^2(H_{\bar{t}}+\alpha {\bar{\Phi}}_{\bar{x}} H_{\bar{x}}) \\ \\
&{\bar{\Phi}}_{\bar{t}} + \frac{1}{2}\alpha ({{\bar{\Phi}}_{\bar{x}}}^2 + \epsilon^2 {{\bar{\Phi}}_{\bar{y}}}^2) + H &= &(B(t)-gh) / ag \\ \\
&{\bar{\Phi}}_{\bar{y}} = 0 &, &\bar{y} = 0
\end{matrix}
</math></center>

where <math>\epsilon = \frac{h}{\lambda}</math> and <math>\alpha=\frac{a}{h}</math> are two small parameters which are given in this problem.

In the next step we use the transform <math>\bar{\Phi} \to \bar{\Phi} + \int\limits_{0}^{\bar{t}}(\frac{B(s) - gh}{ag})ds</math> and introduce further transformation to remove <math>\epsilon</math> from the equations,

<center><math>
z = \frac{\alpha^{1 / 2}}{\epsilon}(\bar{x}-\bar{t}), \quad \tau = \frac{\alpha^{3/2}}{\epsilon}\bar{t}, \quad \Psi = \frac{\alpha^{1/2}}{\epsilon}\bar{\Phi}
</math></center>

The key idea is that <math>\frac{\alpha^{1 / 2}}{\epsilon}</math> is <math>O(1)</math>.

Hence,

<center><math>
\begin{matrix}
&\alpha \Psi_{zz} + \Psi_{\bar{y}\bar{y}} = 0 & -\infin < z <\infin , 0 \le \bar{y} \le 1 + \alpha H(z,\tau) &(1) \\ \\
&\Psi_{\bar{y}} = \alpha (-H_z+\alpha H_{\tau} + \alpha \Psi_z H_z) & y=1+ \alpha H(z,\tau) &(2) \\ \\
&H - \Psi_z + \alpha \Psi_{\tau} + \frac{1}{2} ({\Psi_{\bar{y}}}^2+\alpha {\Psi_z}^2)=0 &y=1+ \alpha H(z,\tau) &(3) \\ \\
&\Psi_{\bar{y}} = 0 &\bar{y}=0 &(4)
\end{matrix}
</math></center>

The boundary condition (4) expresses <math>\Psi</math> at the flat bed, <math>\bar{y}=0</math>. The boundary condition (3) is Bernoulli equation and (2) is kinematic boundary condition. Now we use asymptotic expansions of the form,

<center><math>
\begin{matrix}
&\Psi &= &\Psi_0 + \alpha \Psi_1 + {\alpha}^2 \Psi_2 + o({\alpha}^2) &(5)\\ \\
&H &= &H_0 + \alpha H_1 + o(\alpha) &(6)
\end{matrix}
</math></center>

to derive an equation for each <math>H_i</math> according to the boundary conditions (2) to (4).

* Derivation of <math>H_i</math>'s:

Substituting (5) and (6), (1) must be true for all powers of <math>\alpha</math>. Therefore,

<center><math>
\begin{matrix}
&O(\alpha^0) &: &\Psi_{0, \bar{y}\bar{y}} = 0 &\rArr &\Psi_0 = B_0(z, \tau) \\ \\
&O(\alpha) &: &\Psi_{1, \bar{y}\bar{y}} = -\Psi_{0, zz} &\rArr &\Psi_1 = -\frac{1}{2}{\bar{y}}^2 B_{0, zz}+B_1(z, \tau) \\ \\
&O(\alpha^2) &: &\Psi_{2, \bar{y}\bar{y}} = -\Psi_{1, zz} &\rArr &\Psi_2 = \frac{1}{24}{\bar{y}}^4B_{0,zzzz}-\frac{1}{2}{\bar{y}}^2 B_{1,zz}+ B_2(z, \tau)
\end{matrix}
</math></center>

Now at leading order the Bernoulli and kinematic equations, (3) and (2), gives,

<center><math>
\begin{matrix}
&H_0(z,\tau) = \Psi_{0,z} = B_{0,z} &(a) \\ \\
&H_1-B_{1,z}+\frac{1}{2}B_{0,zzz}+B_{0,\tau}+\frac{1}{2}B^2_{0,z} = 0 &(b) \\ \\
&-H_0B_{0,zz}+\frac{1}{6}B_{0,zzzz}-B_{1,zz} = -H_{1,z}+H_{0,\tau}+B_{0,z}H_{0,z} &(c)
\end{matrix}
</math></center>

== Summery ==
[[Category:789]]

KdV Equation Derivation

2008-10-14T14:32:14Z

Mala058:

KdV Equation Derivation

2008-10-14T12:07:17Z

Mala058:

KdV Equation Derivation

2008-10-14T11:56:26Z

Mala058:

KdV Equation Derivation

2008-10-14T11:39:55Z

Mala058:

KdV Equation Derivation

2008-10-14T11:30:36Z

Mala058:

KdV Equation Derivation

2008-10-14T11:03:26Z

Mala058:

KdV Equation Derivation

2008-10-14T02:48:20Z

Mala058:

[[Category:789]]
We consider the method of derivation of KdV Equation in the concept of [http://www.wikiwaves.org/index.php/Nonlinear_Shallow_Water_Waves Nonlinear Shallow Water Waves].

===Introduction===

In the analysis of [http://www.wikiwaves.org/index.php/Nonlinear_Shallow_Water_Waves Nonlinear Shallow Water Waves] equations we see that there are two important geometrical parameters, <math>\epsilon = \frac{h}{\lambda}</math> and <math>\alpha=\frac{a}{h}</math>, are involved. By choosing appropriate magnitudes for <math>\epsilon</math> and <math>\alpha</math>, we can consider a theory in which dispersion and nonlinearity are in balance. The Korteweg-de Vries Equation verifies the relation between dispersion and nonlinearity properties.