Difference between revisions of "KdV Equation Derivation"

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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].
 
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===
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==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 <b>Korteweg-de Vries Equation</b> verifies the relation between dispersion and nonlinearity properties.
+
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 <b>Korteweg-de Vries Equation</b> verifies the relation between dispersion and nonlinearity properties.
  
===Derivation===
+
==Derivation==
  
 
We begin with the equations for waves on water,
 
We begin with the equations for waves on water,
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To make these equations dimensionless, we use the scaled variables,
 
To make these equations dimensionless, we use the scaled variables,
 
<center><math>
 
<center><math>
\bar{x}=\frac{x}{\lambda}
+
\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}
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</math></center>
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where <math>\sqrt{gh}</math> is defined as <i>linear wave speed in shallow water</i>. Hence the dimensionless system is,
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<center><math>
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\begin{matrix}
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&\epsilon^2 {\bar{\Phi}}_{\bar{x}\bar{x}} + {\bar{\Phi}}_{\bar{y}\bar{y}} &= &0 \\ \\
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&{\bar{\Phi}}_{\bar{y}} &= &\epsilon^2(H_{\bar{t}}+\alpha {\bar{\Phi}}_{\bar{x}} H_{\bar{x}}) \\ \\
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&{\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 \\ \\
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&{\bar{\Phi}}_{\bar{y}} = 0 &, &\bar{y} = 0
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\end{matrix}
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</math></center>
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 +
where <math>\epsilon = \frac{h}{\lambda}</math> and <math>\alpha=\frac{a}{h}</math> are two small parameters which are given in this problem.
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In the next step we use the transform <math>\bar{\Phi} \to \bar{\Phi} + \int\limits_{0}^{\bar{t}}(\frac{B(s) - gh}{ag})\mathrm{d}s</math> and introduce further transformation to remove <font size='3'><math>\epsilon</math></font> from the equations,
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 +
<center><math>
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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}
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</math></center>
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The key idea is that <math>\frac{\alpha^{1 / 2}}{\epsilon}</math> is <font size='3'><math>O(1)</math></font>.
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Hence,
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<center><math>
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\begin{matrix}
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&\alpha \Psi_{zz} + \Psi_{\bar{y}\bar{y}} = 0 & -\infin < z <\infin , 0 \le \bar{y} \le 1 + \alpha H(z,\tau) &(1) \\ \\
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&\Psi_{\bar{y}} = \alpha (-H_z+\alpha H_{\tau} + \alpha \Psi_z H_z) & y=1+ \alpha H(z,\tau) &(2) \\ \\
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&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) \\ \\
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&\Psi_{\bar{y}} = 0 &\bar{y}=0 &(4)
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\end{matrix}
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</math></center>
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The boundary condition (4) expresses <math>\Psi</math> at the flat bed, <math>\bar{y}=0</math>. The boundary condition (3) is <i> Bernoulli equation</i> and (2) is <i>kinematic </i> boundary condition. Now we use asymptotic expansions of the form,
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<center><math>
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\begin{matrix}
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&\Psi &= &\Psi_0 + \alpha \Psi_1 + {\alpha}^2 \Psi_2 + o({\alpha}^3)  &(5)\\ \\
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&H &= &H_0 + \alpha H_1 + o(\alpha^2) &(6)
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\end{matrix}
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</math></center>
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 +
to derive an equation for each <font size='3'><math>H_i</math></font> according to the boundary conditions (2) to (4).
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 +
 
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<b>* Derivation of <font size='3'><math>H_i</math></font>'s:</b>
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 +
Substituting (5) and (6), (1) must be true for all powers of <font size='3'><math>\alpha</math></font>. Therefore,
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 +
<center><math>
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\begin{matrix}
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&O(\alpha^0) &: &\Psi_{0, \bar{y}\bar{y}} = 0  &\rArr &\Psi_0 = B_0(z, \tau) \\ \\
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&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) \\ \\
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&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)
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\end{matrix}
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</math></center>
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Now at leading order the Bernoulli and kinematic equations, (3) and (2), gives,
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<center><math>
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\begin{matrix}
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&H_0(z,\tau) = \Psi_{0,z} = B_{0,z} &(a) \\ \\
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&H_1-B_{1,z}+\frac{1}{2}B_{0,zzz}+B_{0,\tau}+\frac{1}{2}B^2_{0,z} = 0 &(b) \\ \\
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&-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)
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\end{matrix}
 
</math></center>
 
</math></center>
  
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Differentiating (b) and eliminating <math>H_1</math> and <math>B_1</math> from (c) allow us to write,
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 +
<center><math>
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-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}
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</math></center>
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Finally, (a) gives <math>B_0</math> in terms of <math>H_0</math> and hence
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<center><math>
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2H_{0,\tau}+3H_0H_{0,z}+\frac{1}{3}H_{0,zzz}=0
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</math></center>
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which is named <b>Korteweg-de Vries (KdV)</b> equation.
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==Interpretation==
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KdV equation includes dispersive effects through the term <font size='3'><math>H_{0,zzz}</math></font> and nonlinear effects through the term <font size='3'><math>H_0H_{0,z}</math></font> and governs the behavior of the small amplitude waves, with <font size='3'><math>\alpha<<1</math></font>. It is reasonable to ask when and where the independent variables, <font size='3'><math>z</math></font> and <font size='3'><math>\tau</math></font>, are of <font size='3'><math>O(1)</math></font> 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 <font size='3'><math>z</math></font> and <font size='3'><math>\tau</math></font>, if <font size='3'><math>\alpha=O(\epsilon^2)</math></font>, 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.
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For solution of KdV equation please refer [http://www.wikiwaves.org/index.php/KdV_Equation_Solutions here.]
  
===Summery===
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[[Category:Nonlinear Water-Wave Theory]]
[[Category:789]]
 

Latest revision as of 09:56, 6 November 2010



We consider the method of derivation of KdV Equation in the concept of Nonlinear Shallow Water Waves.

Introduction

In the analysis of Nonlinear Shallow Water Waves equations we see that there are two important geometrical parameters, [math]\displaystyle{ \epsilon = \frac{h}{\lambda} }[/math] and [math]\displaystyle{ \alpha=\frac{a}{h} }[/math] involved. By choosing appropriate magnitudes for [math]\displaystyle{ \epsilon }[/math] and [math]\displaystyle{ \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,

[math]\displaystyle{ \begin{matrix} &\Phi_{xx} + \Phi_{yy} &= 0 \quad &-\infin\lt x\lt \infin, 0 \le y \le \eta(x,t) \\ \end{matrix} }[/math]

Provided that at [math]\displaystyle{ y=\eta(x,t)=h+aH(x,t) }[/math] we have,

[math]\displaystyle{ \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]

To make these equations dimensionless, we use the scaled variables,

[math]\displaystyle{ \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]

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

[math]\displaystyle{ \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]

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

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

[math]\displaystyle{ 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]

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

Hence,

[math]\displaystyle{ \begin{matrix} &\alpha \Psi_{zz} + \Psi_{\bar{y}\bar{y}} = 0 & -\infin \lt z \lt \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]

The boundary condition (4) expresses [math]\displaystyle{ \Psi }[/math] at the flat bed, [math]\displaystyle{ \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,

[math]\displaystyle{ \begin{matrix} &\Psi &= &\Psi_0 + \alpha \Psi_1 + {\alpha}^2 \Psi_2 + o({\alpha}^3) &(5)\\ \\ &H &= &H_0 + \alpha H_1 + o(\alpha^2) &(6) \end{matrix} }[/math]

to derive an equation for each [math]\displaystyle{ H_i }[/math] according to the boundary conditions (2) to (4).


* Derivation of [math]\displaystyle{ H_i }[/math]'s:

Substituting (5) and (6), (1) must be true for all powers of [math]\displaystyle{ \alpha }[/math]. Therefore,

[math]\displaystyle{ \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]

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

[math]\displaystyle{ \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]

Differentiating (b) and eliminating [math]\displaystyle{ H_1 }[/math] and [math]\displaystyle{ B_1 }[/math] from (c) allow us to write,

[math]\displaystyle{ -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]

Finally, (a) gives [math]\displaystyle{ B_0 }[/math] in terms of [math]\displaystyle{ H_0 }[/math] and hence

[math]\displaystyle{ 2H_{0,\tau}+3H_0H_{0,z}+\frac{1}{3}H_{0,zzz}=0 }[/math]

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

Interpretation

KdV equation includes dispersive effects through the term [math]\displaystyle{ H_{0,zzz} }[/math] and nonlinear effects through the term [math]\displaystyle{ H_0H_{0,z} }[/math] and governs the behavior of the small amplitude waves, with [math]\displaystyle{ \alpha\lt \lt 1 }[/math]. It is reasonable to ask when and where the independent variables, [math]\displaystyle{ z }[/math] and [math]\displaystyle{ \tau }[/math], are of [math]\displaystyle{ 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]\displaystyle{ z }[/math] and [math]\displaystyle{ \tau }[/math], if [math]\displaystyle{ \alpha=O(\epsilon^2) }[/math], then [math]\displaystyle{ \bar{t}\gt \gt 1 }[/math] and [math]\displaystyle{ \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 here.