Miura transform: Difference between revisions
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In the defocusing case, the ''Miura transformation'' <math> v = \partial_x u + u^2 </math> transforms a solution of defocussing [[mKdV]] to a solution of [[KdV]] | |||
In the defocusing case, the ''Miura transformation'' <math> v = \partial_x u + u^2 </math> transforms a solution of | |||
<center><math>\partial_t v + \partial_x^3 v = 6 v \partial_x v</math>.</center> | <center><math>\partial_t v + \partial_x^3 v = 6 v \partial_x v</math>.</center> | ||
Thus one expects the LWP and GWP theory for | Thus one expects the LWP and GWP theory for mKdV to be one derivative higher than that for KdV. | ||
In the focusing case, the Miura transform is now <math>v = \partial_x u + i u^2</math>. This transforms <span class="SpellE">focussing</span | In the focusing case, the Miura transform is now <math>v = \partial_x u + i u^2</math>. This transforms <span class="SpellE">focussing</span> mKdV to ''complex-valued'' KdV, which is a slightly less tractable equation. (However, the transformed solution v is still real in the highest order term, so in principle the real-valued theory carries over to this case. | ||
The Miura transformation can be generalized. If v and w solve the system | The Miura transformation can be generalized. If v and w solve the system | ||
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<span class="GramE">then</span> <math>u = a^2 v^2 + a \partial_x v + bv</math> solves <span class="SpellE">KdV</span> (this is the ''Gardener transform''). | <span class="GramE">then</span> <math>u = a^2 v^2 + a \partial_x v + bv</math> solves <span class="SpellE">KdV</span> (this is the ''Gardener transform''). | ||
[[Category:Integrability]] | |||
[[Category:Airy]] | |||
[[Category:Transforms]] | |||
Latest revision as of 07:40, 31 July 2006
In the defocusing case, the Miura transformation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle v = \partial_x u + u^2 } transforms a solution of defocussing mKdV to a solution of KdV
Thus one expects the LWP and GWP theory for mKdV to be one derivative higher than that for KdV.
In the focusing case, the Miura transform is now Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle v = \partial_x u + i u^2} . This transforms focussing mKdV to complex-valued KdV, which is a slightly less tractable equation. (However, the transformed solution v is still real in the highest order term, so in principle the real-valued theory carries over to this case.
The Miura transformation can be generalized. If v and w solve the system
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \partial_t w + \partial_x^3 w = 6(v^2 + w) \partial_x w}
Then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u = v^2 + \partial_x v + w } is a solution of KdV. In particular, if a and b are constants and v solves
then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u = a^2 v^2 + a \partial_x v + bv} solves KdV (this is the Gardener transform).
