Published in:Communications in Theoretical Physics
Language:
Not determined
DOI:
10.1088/1572-9494/acce97
ISSN:
0253-6102;
1572-9494
Origination:
Footnote:
Description:
<jats:title>Abstract</jats:title>
<jats:p>In this paper, we investigate the fifth-order modified Korteweg–de Vries (mKdV) equation on the half-line via the Fokas unified transformation approach. We show that the solution <jats:italic>u</jats:italic>(<jats:italic>x</jats:italic>, <jats:italic>t</jats:italic>) of the fifth-order mKdV equation can be represented by the solution of the matrix Riemann-Hilbert problem constructed on the plane of complex spectral parameter <jats:italic>θ</jats:italic>. The jump matrix <jats:italic>L</jats:italic>(<jats:italic>x</jats:italic>, <jats:italic>t</jats:italic>, <jats:italic>θ</jats:italic>) has an explicit representation dependent on <jats:italic>x</jats:italic>, <jats:italic>t</jats:italic> and it can be represented exactly by the two pairs of spectral functions <jats:italic>y</jats:italic>(<jats:italic>θ</jats:italic>), <jats:italic>z</jats:italic>(<jats:italic>θ</jats:italic>) (obtained from the initial value <jats:italic>u</jats:italic>
<jats:sub>0</jats:sub>(<jats:italic>x</jats:italic>)) and <jats:italic>Y</jats:italic>(<jats:italic>θ</jats:italic>), <jats:italic>Z</jats:italic>(<jats:italic>θ</jats:italic>) (obtained from the boundary conditions <jats:italic>v</jats:italic>
<jats:sub>0</jats:sub>(<jats:italic>t</jats:italic>), <jats:inline-formula>
<jats:tex-math>
<?CDATA ${\{{v}_{k}(t)\}}_{1}^{4}$?>
</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll">
<mml:msubsup>
<mml:mrow>
<mml:mo stretchy="false">{</mml:mo>
<mml:msub>
<mml:mrow>
<mml:mi>v</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>t</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo stretchy="false">}</mml:mo>
</mml:mrow>
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mn>4</mml:mn>
</mml:mrow>
</mml:msubsup>
</mml:math>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="ctpacce97ieqn1.gif" xlink:type="simple" />
</jats:inline-formula>). Furthermore, the two pairs of spectral functions <jats:italic>y</jats:italic>(<jats:italic>θ</jats:italic>), <jats:italic>z</jats:italic>(<jats:italic>θ</jats:italic>) and <jats:italic>Y</jats:italic>(<jats:italic>θ</jats:italic>), <jats:italic>Z</jats:italic>(<jats:italic>θ</jats:italic>) are not independent of each other, but are related to the compatibility condition, the so-called global relation.</jats:p>