Generalized Hyperbolic Function Solution to a Class of Nonlinear Schrödinger-Type Equations

<table>Evolutional behaviour of the intensity <svg height="19.924999" id="M162" style="vertical-align:-3.276pt" version="1.1" viewbox="0 0 34.849998 19.924999" width="34.849998" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,15.775)"><path d="M162 -163h-61v866h61v-866z" id="x7C"></path></g><g transform="matrix(.017,-0,0,-.017,4.533,15.775)"><use xlink:href="#x1D462"></use></g>
<g transform="matrix(.012,-0,0,-.012,13.675,19.863)"><use xlink:href="#x31"></use></g>
<g transform="matrix(.017,-0,0,-.017,20.012,15.775)"><use xlink:href="#x7C"></use></g>
<g transform="matrix(.012,-0,0,-.012,24.488,7.613)"><use xlink:href="#x32"></use></g>
</svg>describes the bright soliton:  (a) <svg height="15.2875" id="M163" style="vertical-align:-3.50804pt" version="1.1" viewbox="0 0 89.962502 15.2875" width="89.962502" 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)"><use xlink:href="#x1D45D"></use></g><g transform="matrix(.017,-0,0,-.017,14.851,10.862)"><use xlink:href="#x3D"></use></g><g transform="matrix(.017,-0,0,-.017,29.555,10.862)"><use xlink:href="#x31"></use></g><g transform="matrix(.017,-0,0,-.017,37.715,10.862)"><use xlink:href="#x2C"></use></g><g transform="matrix(.017,-0,0,-.017,44.429,10.862)"><use xlink:href="#x1D45E"></use></g><g transform="matrix(.017,-0,0,-.017,57.484,10.862)"><use xlink:href="#x3D"></use></g><g transform="matrix(.017,-0,0,-.017,72.188,10.862)"><use xlink:href="#x31"></use></g><g transform="matrix(.017,-0,0,-.017,80.348,10.862)"><use xlink:href="#x3B"></use></g>
</svg>(b) <svg height="16.4375" id="M164" style="vertical-align:-3.50804pt" version="1.1" viewbox="0 0 143.3875 16.4375" width="143.3875" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.013)"><use xlink:href="#x1D45D"></use></g><g transform="matrix(.017,-0,0,-.017,14.851,12.013)"><use xlink:href="#x3D"></use></g><g transform="matrix(.017,-0,0,-.017,29.555,12.013)"><use xlink:href="#x31"></use></g><g transform="matrix(.017,-0,0,-.017,37.715,12.013)"><use xlink:href="#x2F"></use></g><g transform="matrix(.017,-0,0,-.017,44.718,12.013)"><use xlink:href="#x33"></use></g><g transform="matrix(.017,-0,0,-.017,52.878,12.013)"><use xlink:href="#x33"></use></g><g transform="matrix(.017,-0,0,-.017,61.037,12.013)"><use xlink:href="#x32"></use></g><g transform="matrix(.017,-0,0,-.017,69.196,12.013)"><use xlink:href="#x2C"></use></g><g transform="matrix(.017,-0,0,-.017,75.911,12.013)"><use xlink:href="#x1D45E"></use></g><g transform="matrix(.017,-0,0,-.017,88.966,12.013)"><use xlink:href="#x3D"></use></g><g transform="matrix(.017,-0,0,-.017,103.67,12.013)"><use xlink:href="#x31"></use></g><g transform="matrix(.017,-0,0,-.017,111.829,12.013)"><use xlink:href="#x2F"></use></g><g transform="matrix(.017,-0,0,-.017,118.833,12.013)"><use xlink:href="#x33"></use></g><g transform="matrix(.017,-0,0,-.017,126.992,12.013)"><use xlink:href="#x33"></use></g><g transform="matrix(.017,-0,0,-.017,135.152,12.013)"><use xlink:href="#x33"></use></g>
</svg> (singular wave solution); (c) <svg height="15.2875" id="M165" style="vertical-align:-3.50804pt" version="1.1" viewbox="0 0 137.125 15.2875" width="137.125" 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)"><use xlink:href="#x1D45D"></use></g><g transform="matrix(.017,-0,0,-.017,14.851,10.862)"><use xlink:href="#x3D"></use></g><g transform="matrix(.017,-0,0,-.017,29.555,10.862)"><use xlink:href="#x32"></use></g><g transform="matrix(.017,-0,0,-.017,37.715,10.862)"><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,-.017,41.59,10.862)"><use xlink:href="#x31"></use></g><g transform="matrix(.017,-0,0,-.017,49.75,10.862)"><use xlink:href="#x33"></use></g><g transform="matrix(.017,-0,0,-.017,57.909,10.862)"><use xlink:href="#x32"></use></g><g transform="matrix(.017,-0,0,-.017,66.069,10.862)"><use xlink:href="#x2C"></use></g><g transform="matrix(.017,-0,0,-.017,72.783,10.862)"><use xlink:href="#x1D45E"></use></g><g transform="matrix(.017,-0,0,-.017,85.838,10.862)"><use xlink:href="#x3D"></use></g><g transform="matrix(.017,-0,0,-.017,100.542,10.862)"><use xlink:href="#x31"></use></g><g transform="matrix(.017,-0,0,-.017,108.702,10.862)"><use xlink:href="#x2E"></use></g><g transform="matrix(.017,-0,0,-.017,112.577,10.862)"><use xlink:href="#x37"></use></g><g transform="matrix(.017,-0,0,-.017,120.737,10.862)"><use xlink:href="#x32"></use></g><g transform="matrix(.017,-0,0,-.017,128.896,10.862)"><use xlink:href="#x33"></use></g>
</svg>.</table>

Journal of Applied Mathematics

fig7

Figure 7

Figure 7: Generalized Hyperbolic Function Solution to a Class of Nonlinear Schrödinger-Type Equations 