Korteweg-de Vries equation on R: Difference between revisions

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The <span class="SpellE">KdV</span> equation can also be generalized to a 2x2 system
The <span class="SpellE">KdV</span> equation can also be generalized to a 2x2 system


<center><span class="SpellE"><tt><font size="10.0pt">u_t</font></tt></span><tt><font size="10.0pt"> + <span class="SpellE">u_xxx</span> + a_3 <span class="SpellE">v_xxx</span> + u <span class="SpellE">u_x</span> + a_1 v <span class="SpellE">v_x</span> + a_2 (<span class="SpellE">uv</span><span class="GramE">)_</span>x = 0</font></tt><br /><tt><font size="10.0pt">b_1 <span class="SpellE">v_t</span> + <span class="SpellE">v_xxx</span> + b_2 a_3 <span class="SpellE">u_xxx</span> + v <span class="SpellE">v_x</span> + b_2 a_2 u <span class="SpellE">u_x</span> + b_2 a_1 (<span class="SpellE">uv</span>)_x + r <span class="SpellE">v_x</span></font></tt></center>
<center><math>\partial_t u + \partial_x^3 u + a_3 \partial_x^3 v + u \partial_x u + a_1 v \partial_x v + a_2 \partial_x (uv)= 0<br />b_1 \partial_t v + \partial_x^3 v + b_2 a_3 \partial_x^3 u + v \partial_x v + b_2 a_2 u \partial_x u + b_2 a_1 \partial_x (uv) + r \partial_x v</math></center>


<span class="GramE">where</span> b_1,b_2 are positive constants and a_1,a_2,a_3,r are real constants. This system was introduced in [[references.html#GeaGr1984 GeaGr1984]] to study strongly interacting pairs of weakly nonlinear long waves, and studied further in [[references.html#BnPoSauTm1992 BnPoSauTm1992]]. In [[references.html#AsCoeWgg1996 AsCoeWgg1996]] it was shown that this system was also globally well-posed on L^2. <br /> It is an interesting question as to whether these results can be pushed further to match the <span class="SpellE">KdV</span> theory; the apparent lack of complete <span class="SpellE">integrability</span> in this system (for generic choices of parameters <span class="SpellE">b_i</span>, <span class="SpellE">a_i</span>, <span class="GramE">r</span>) suggests a possible difficulty.
<span class="GramE">where</span> b_1,b_2 are positive constants and a_1,a_2,a_3,r are real constants. This system was introduced in [[references.html#GeaGr1984 GeaGr1984]] to study strongly interacting pairs of weakly nonlinear long waves, and studied further in [[references.html#BnPoSauTm1992 BnPoSauTm1992]]. In [[references.html#AsCoeWgg1996 AsCoeWgg1996]] it was shown that this system was also globally well-posed on L^2. <br /> It is an interesting question as to whether these results can be pushed further to match the <span class="SpellE">KdV</span> theory; the apparent lack of complete <span class="SpellE">integrability</span> in this system (for generic choices of parameters <span class="SpellE">b_i</span>, <span class="SpellE">a_i</span>, <span class="GramE">r</span>) suggests a possible difficulty.


[[Category:Equations]]
[[Category:Equations]]

Revision as of 20:15, 28 July 2006

The local and global well-posedness theory for the Korteweg-de Vries equation on the real line is as follows.

The KdV equation can also be generalized to a 2x2 system

where b_1,b_2 are positive constants and a_1,a_2,a_3,r are real constants. This system was introduced in references.html#GeaGr1984 GeaGr1984 to study strongly interacting pairs of weakly nonlinear long waves, and studied further in references.html#BnPoSauTm1992 BnPoSauTm1992. In references.html#AsCoeWgg1996 AsCoeWgg1996 it was shown that this system was also globally well-posed on L^2.
It is an interesting question as to whether these results can be pushed further to match the KdV theory; the apparent lack of complete integrability in this system (for generic choices of parameters b_i, a_i, r) suggests a possible difficulty.