Trigonometric Function Periodic Wave Solutions and Their Limit Forms for the KdV-Like and the PC-Like Equations

<table>When <svg height="17.674999" id="M422" style="vertical-align:-3.13504pt" version="1.1" viewbox="0 0 46.8125 17.674999" width="46.8125" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,17.675)">
<g transform="translate(72,-57.86)">
<text transform="matrix(1,0,0,-1,-71.95,61.03)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑐</tspan>
</text>
<text transform="matrix(1,0,0,-1,-66.14,66.03)">
<tspan style="font-size: 8.75px; " x="0" y="0">2</tspan>
</text>
<text transform="matrix(1,0,0,-1,-57.79,61.03)">
<tspan style="font-size: 12.50px; " x="0" y="0">&lt;</tspan>
<tspan style="font-size: 12.50px; " x="12.027886" y="0">𝑎</tspan>
</text>
<text transform="matrix(1,0,0,-1,-39.47,57.91)">
<tspan style="font-size: 8.75px; " x="0" y="0">1</tspan>
</text>
</g>
</g>
</svg>, the bifurcation phase portraits of system (<a href="https://static.hindawi.com/articles/mpe/volume-2011/810217/figures/#EEq5.3">5.3</a>) and the locations of <svg height="11.05" id="M423" style="vertical-align:-3.2316pt" version="1.1" viewbox="0 0 15.25 11.05" width="15.25" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,11.05)">
<g transform="translate(72,-63.16)">
<text transform="matrix(1,0,0,-1,-71.95,66.44)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑚</tspan>
</text>
<text transform="matrix(1,0,0,-1,-63.07,63.31)">
<tspan style="font-size: 8.75px; " x="0" y="0">𝑖</tspan>
</text>
</g>
</g>
</svg> and <svg height="14.7125" id="M424" style="vertical-align:-3.2316pt" version="1.1" viewbox="0 0 88.25 14.7125" width="88.25" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,14.7125)">
<g transform="translate(72,-60.23)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑛</tspan>
</text>
<text transform="matrix(1,0,0,-1,-65.74,60.37)">
<tspan style="font-size: 8.75px; " x="0" y="0">𝑖</tspan>
</text>
<text transform="matrix(1,0,0,-1,-58.35,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">(</tspan>
<tspan style="font-size: 12.50px; " x="4.1634989" y="0">𝑖</tspan>
<tspan style="font-size: 12.50px; " x="11.515263" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="23.555653" y="0">1</tspan>
<tspan style="font-size: 12.50px; " x="29.807152" y="0">,</tspan>
<tspan style="font-size: 12.50px; " x="35.020905" y="0">2</tspan>
<tspan style="font-size: 12.50px; " x="41.272404" y="0">,</tspan>
<tspan style="font-size: 12.50px; " x="46.486153" y="0">3</tspan>
<tspan style="font-size: 12.50px; " x="52.737656" y="0">)</tspan>
</text>
</g>
</g>
</svg>.</table>

Mathematical Problems in Engineering

fig10

Figure 10

Figure 10: Trigonometric Function Periodic Wave Solutions and Their Limit Forms for the KdV-Like and the PC-Like Equations 