Different Kinds of Singular and Nonsingular Exact Traveling Wave Solutions of the Kudryashov-Sinelshchikov Equation in the Special Parametric Conditions

<table class="table-group" width="800px"><tr class="fig-image" id="a"><td><img alt="456964.fig.008a" src="https://static.hindawi.com/articles/mpe/volume-2013/456964/figures/456964.fig.008a.jpg"/></td></tr><tr class="fig-caption"><td align="center"><table><tr><td><b>(a) </b>Generalized kink wave</td></tr></table></td></tr><tr class="fig-image" id="b"><td><img alt="456964.fig.008b" src="https://static.hindawi.com/articles/mpe/volume-2013/456964/figures/456964.fig.008b.jpg"/></td></tr><tr class="fig-caption"><td align="center"><table><tr><td><b>(b) </b>Generalized kink wave</td></tr></table></td></tr></table>

<table>The profiles of soliton solutions (<a href="https://static.hindawi.com/articles/mpe/volume-2013/456964/figures/#EEq3.20">32</a>) for (a)  <svg height="14.2125" id="M169" style="vertical-align:-1.59705pt" version="1.1" viewbox="0 0 98.737503 14.2125" width="98.737503" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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q-14 98 -31 149.5t-50 51.5q-49 0 -94 -70t-45 -164q0 -55 15.5 -86t40.5 -31q70 0 164 150z" id="x1D6FC"></path></g><g transform="matrix(.017,-0,0,-.017,14.426,12.162)"><path d="M535 323h-483v50h483v-50zM535 138h-483v50h483v-50z" id="x3D"></path></g><g transform="matrix(.017,-0,0,-.017,29.13,12.162)"><path d="M241 635q53 0 94 -28.5t63.5 -76t33.5 -102.5t11 -116q0 -58 -11 -112.5t-34 -103.5t-63.5 -78.5t-94.5 -29.5t-95 28t-64.5 75t-34.5 102.5t-11 118.5q0 58 11.5 112.5t34.5 103t64.5 78t95.5 29.5zM238 602q-32 0 -55.5 -25t-35.5 -68t-17.5 -91t-5.5 -105
q0 -76 10 -138.5t37 -107.5t69 -45q32 0 55.5 25t35.5 68.5t17.5 91.5t5.5 105t-5.5 105.5t-18 92t-36 68t-56.5 24.5z" id="x30"></path></g><g transform="matrix(.017,-0,0,-.017,37.29,12.162)"><path d="M113 -12q-24 0 -39.5 16t-15.5 42q0 24 16 40.5t40 16.5t40 -16.5t16 -40.5q0 -26 -16 -42t-41 -16z" id="x2E"></path></g><g transform="matrix(.017,-0,0,-.017,41.165,12.162)"><path d="M153 550l-26 -186q79 31 111 31q90 0 141.5 -51t51.5 -119q0 -93 -89 -166q-85 -69 -173 -71q-32 0 -61.5 11.5t-41.5 23.5q-18 17 -17 34q2 16 22 33q14 9 26 -1q61 -50 124 -50q60 0 93 43.5t33 104.5q0 69 -41.5 110t-121.5 41q-53 0 -102 -20l38 305h286l6 -8
l-26 -65h-233z" id="x35"></path></g><g transform="matrix(.017,-0,0,-.017,49.325,12.162)"><path d="M95 130q31 0 61 -30t30 -78q0 -53 -38 -87.5t-93 -51.5l-11 29q77 31 77 85q0 26 -17.5 43t-44.5 24q-4 0 -8.5 6.5t-4.5 17.5q0 18 15 30t34 12z" id="x2C"></path></g><g transform="matrix(.017,-0,0,-.017,61.7,12.162)"><path d="M529 97q-70 -109 -136 -109q-41 0 -56 94q-23 144 -37 284q-38 -88 -99 -202.5t-93 -156.5q-26 -8 -76 -19l-9 21q71 78 145.5 193t124.5 232q-5 84 -15 128q-12 55 -29.5 75.5t-42.5 20.5q-21 0 -45 -13l-8 24q16 17 46 30t55 13q43 0 70 -46.5t40 -169.5
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</svg>,  and <svg height="11.125" id="M171" style="vertical-align:-0.1638pt" version="1.1" viewbox="0 0 44.887501 11.125" width="44.887501" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,10.862)"><path d="M401 397q0 -24 -24 -40q-18 -13 -31 3q-36 45 -93 45q-48 0 -86.5 -43t-51.5 -112l206 6l-8 -48l-203 6v-25q0 -68 26.5 -107t73.5 -39q73 0 148 64l16 -22q-40 -44 -91 -70.5t-102 -26.5q-75 0 -116.5 45t-41.5 136q0 125 99 208q36 30 87 50.5t93 20.5q46 0 72.5 -16
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</svg> and (b)  <svg height="14.2125" id="M172" style="vertical-align:-1.59705pt" version="1.1" viewbox="0 0 168.28751 14.2125" width="168.28751" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.162)"><path d="M545 106q-67 -118 -134 -118q-24 0 -40 37.5t-30 129.5h-2q-47 -72 -103 -119.5t-108 -47.5q-47 0 -76 45.5t-29 119.5q0 113 85 204t174 91q47 0 70 -33.5t43 -119.5h3q32 47 80 140l55 13l10 -9q-47 -80 -138 -201q17 -99 27.5 -136t22.5 -37q23 0 69 61zM333 204
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</svg>, and <svg height="11.125" id="M173" style="vertical-align:-0.1638pt" version="1.1" viewbox="0 0 34.912498 11.125" width="34.912498" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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</svg>. </table>

Mathematical Problems in Engineering

fig8

Figure 8

Figure 8: Different Kinds of Singular and Nonsingular Exact Traveling Wave Solutions of the Kudryashov-Sinelshchikov Equation in the Special Parametric Conditions 

Figure 8 | Different Kinds of Singular and Nonsingular Exact Traveling Wave Solutions of the Kudryashov-Sinelshchikov Equation in the Special Parametric Conditions 