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

<table>The locations of the rays <svg height="14.5" id="M202" style="vertical-align:-3.2316pt" version="1.1" viewbox="0 0 13.325 14.5" width="13.325" xmlns="http://www.w3.org/2000/svg">
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<tspan style="font-size: 12.50px; " x="0" y="0">Γ</tspan>
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<text transform="matrix(1,0,0,-1,-64.61,60.55)">
<tspan style="font-size: 8.75px; " x="0" y="0">𝑖</tspan>
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</svg>, <svg height="14.3625" id="M203" style="vertical-align:-3.2316pt" version="1.1" viewbox="0 0 15.2125 14.3625" width="15.2125" xmlns="http://www.w3.org/2000/svg">
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<g transform="translate(72,-60.51)">
<text transform="matrix(1,0,0,-1,-71.95,63.79)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝐿</tspan>
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<text transform="matrix(1,0,0,-1,-63.1,60.66)">
<tspan style="font-size: 8.75px; " x="0" y="0">𝑖</tspan>
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</svg> and the regions <svg height="14.3625" id="M204" style="vertical-align:-3.2316pt" version="1.1" viewbox="0 0 18.4625 14.3625" width="18.4625" xmlns="http://www.w3.org/2000/svg">
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<g transform="translate(72,-60.51)">
<text transform="matrix(1,0,0,-1,-71.95,63.79)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑊</tspan>
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<text transform="matrix(1,0,0,-1,-60.5,60.66)">
<tspan style="font-size: 8.75px; " x="0" y="0">𝑖</tspan>
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</svg>, <svg height="14.7125" id="M205" style="vertical-align:-3.2316pt" version="1.1" viewbox="0 0 116.8625 14.7125" width="116.8625" 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">Ω</tspan>
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<text transform="matrix(1,0,0,-1,-62.65,60.37)">
<tspan style="font-size: 8.75px; " x="0" y="0">𝑖</tspan>
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<text transform="matrix(1,0,0,-1,-55.26,63.5)">
<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="46.473652" y="0">…</tspan>
<tspan style="font-size: 12.50px; " x="61.064651" y="0">,</tspan>
<tspan style="font-size: 12.50px; " x="66.278404" y="0">8</tspan>
<tspan style="font-size: 12.50px; " x="72.5299" y="0">)</tspan>
</text>
</g>
</g>
</svg> for given <svg height="10.925" id="M206" style="vertical-align:-3.13504pt" version="1.1" viewbox="0 0 14.0625 10.925" width="14.0625" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,10.925)">
<g transform="translate(72,-63.26)">
<text transform="matrix(1,0,0,-1,-71.95,66.44)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑎</tspan>
</text>
<text transform="matrix(1,0,0,-1,-65.67,63.31)">
<tspan style="font-size: 8.75px; " x="0" y="0">1</tspan>
</text>
</g>
</g>
</svg> and <svg height="7.1875" id="M207" style="vertical-align:-0.13794pt" version="1.1" viewbox="0 0 7.3874998 7.1875" width="7.3874998" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,7.1875)">
<g transform="translate(72,-66.25)">
<text transform="matrix(1,0,0,-1,-71.95,66.44)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑐</tspan>
</text>
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</g>
</svg>.</table>

Mathematical Problems in Engineering

fig7

Figure 7

Figure 7: Trigonometric Function Periodic Wave Solutions and Their Limit Forms for the KdV-Like and the PC-Like Equations 