Exact Periodic Solutions of the Nonintegrable Kawahara Equation

<table>Real sinusoidal periodic solutions for real values of the wave number: <svg height="12.8875" id="M154" style="vertical-align:-1.76814pt" version="1.1" viewbox="0 0 62.287498 12.8875" width="62.287498" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,12.8875)">
<g transform="translate(72,-61.69)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑘</tspan>
<tspan style="font-size: 12.50px; " x="10.239957" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="22.280346" y="0">1</tspan>
<tspan style="font-size: 12.50px; " x="28.531845" y="0">/</tspan>
<tspan style="font-size: 12.50px; " x="32.007679" y="0">2</tspan>
<tspan style="font-size: 12.50px; " x="38.259178" y="0">,</tspan>
<tspan style="font-size: 12.50px; " x="43.472931" y="0">1</tspan>
</text>
</g>
</g>
</svg> and <svg height="13.9125" id="M155" style="vertical-align:-2.29482pt" version="1.1" viewbox="0 0 48.400002 13.9125" width="48.400002" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,13.9125)">
<g transform="translate(72,-60.87)">
<text transform="matrix(1,0,0,-1,-71.95,63.2)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑞</tspan>
<tspan style="font-size: 12.50px; " x="9.1897049" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="21.230095" y="0">𝑒</tspan>
</text>
<text transform="matrix(1,0,0,-1,-45.15,68.2)">
<tspan style="font-size: 8.75px; " x="0" y="0">−</tspan>
<tspan style="font-size: 8.75px; " x="5.99512" y="0">𝜋</tspan>
</text>
</g>
</g>
</svg>, <svg height="10.8625" id="M156" style="vertical-align:-0.13794pt" version="1.1" viewbox="0 0 34.887501 10.8625" width="34.887501" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,10.8625)">
<g transform="translate(72,-63.31)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝜀</tspan>
<tspan style="font-size: 12.50px; " x="9.5147829" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="21.555172" y="0">1</tspan>
</text>
</g>
</g>
</svg>.</table>

International Scholarly Research Notices

Exact Periodic Solutions of the Nonintegrable Kawahara Equation

Figure 2