Toda Lattice Solitons

<body bgcolor=oldlace> <center><b> Toda Lattice Solitons </b></center> <applet code="TodaApplet.class" width=600 height=400> <param name=minwidthpanel value="300"> <center> <i>Java</i> support is required for this interface. <br> In order to see <i>Toda Lattice Solitons</i> applet<br> you need a <i>Java</i> compatible browser. </center> </applet> <hr> <center><b> Toda Lattice Solitons </b></center> <hr width=75%> <i>Toda lattice</i> is a monatomic chain of particles of mass <i>m</i> with the following interaction between nearest-neighbors <br> <center> <b> <I>V</I>(<I>r</I>) = (<I>a</I>/<I>b</I>)<I>exp</I>(-<I>br</I>)+<I>ar</I>-<I>a</I>/<I>b</I></b> , </center> where <b><i>r</i></b> is the deviation of the nearest-neighbor distance from its equilibrium value, and <nobr><b><i>ab</i> > 0</b></nobr>. <p> Such lattice supports soliton solutions. For a single soliton the deviation of the <i>n</i>th particle from its equilibrium position can be written as <br> <img align=middle src="images/onesol.gif"> , <br> where the lattice constant is chosen as the unit of length, and <img align=middle src="images/betaeql.gif"> . <p> The soliton is a pulslike wave with the speed <img align=middle src="images/speedsol.gif">. It always moves faster than plane waves. (The speed of sound is <img align=middle src="images/speedsnd.gif">.) The smaller the soliton's width (<b>~1/<i>k</i></b>), the faster it propagates. <p> The lattice is compressed by the value <b>2<i>k</i>/<i>b</i></b> around a soliton (we assume <nobr><b><i>b</i> > 0</b></nobr>). Due to such a compression, we can speak of the soliton's <nobr>"<b>mass</b>" = <b>2<i>mk</i>/<i>b</i></b>.</nobr> <p> A more complete description of the Toda solitons you can find in the book: <br> M. Toda, <i>Theory of Nonlinear Lattices</i>, (Springer-Verlag, New York 1989). <hr width=75%> The above <font color=red><b><i>applet</i></b></font> allows you to watch propagating solitons in the lattice of 15 particles with fixed boundaries. The evolution of the chain is calculated by the molecular-dynamics technique. The parameters of the lattice are the following: <nobr><b><i>a</i>=1, <i>b</i>=1, <i>m</i>=1</b></nobr>. <p> You can launch either a <i>wide soliton</i> (<b><i>k</i>=0.5</b>) or a <i>sharp</i> one (<b><i>k</i>=1</b>). You can also see non-destructive collisions of <i>two sharp solitons</i>. <p> The time is shown in units of the shortest period of small amplitude plane wave vibrations, <img align=middle src="images/tmeql.gif">. Energy is shown in arbitrary units. The <font color=red><b>kinetic</b></font> energy of the particle and the <font color="ccaa00"><b>potential</b></font> energy of the bond are shown as the <font color=red><b>red</b></font> and the <font color="#ccaa00"><b>yellow</b></font> bars, respectively. <p> If you wait for a while you will see a power spectrum of the particles' vibrations. It will be shown in the left panel. As the time of the evolution goes the spectrum resolution improves. The frequency unit is the maximal plane wave frequency, <img align=middle src="images/wmeql.gif">. <hr> <font size=-2> Last modified: December 1, 1996<br> <A HREF="http://www.lightlink.com/sergey/">Sergey Kiselev</a>, <i><a href="mailto:sergey@lightlink.com">sergey@lightlink.com</a></i><br> </font> </BODY> <body bgcolor=oldlace> <center><b> Toda Lattice Solitons </b></center> <applet code="TodaApplet.class" width=600 height=400> <param name=minwidthpanel value="300"> <center> <i>Java</i> support is required for this interface. <br> In order to see <i>Toda Lattice Solitons</i> applet<br> you need a <i>Java</i> compatible browser. </center> </applet> <hr> <center><b> Toda Lattice Solitons </b></center> <hr width=75%> <i>Toda lattice</i> is a monatomic chain of particles of mass <i>m</i> with the following interaction between nearest-neighbors <br> <center> <b> <I>V</I>(<I>r</I>) = (<I>a</I>/<I>b</I>)<I>exp</I>(-<I>br</I>)+<I>ar</I>-<I>a</I>/<I>b</I></b> , </center> where <b><i>r</i></b> is the deviation of the nearest-neighbor distance from its equilibrium value, and <nobr><b><i>ab</i> > 0</b></nobr>. <p> Such lattice supports soliton solutions. For a single soliton the deviation of the <i>n</i>th particle from its equilibrium position can be written as <br> <img align=middle src="images/onesol.gif"> , <br> where the lattice constant is chosen as the unit of length, and <img align=middle src="images/betaeql.gif"> . <p> The soliton is a pulslike wave with the speed <img align=middle src="images/speedsol.gif">. It always moves faster than plane waves. (The speed of sound is <img align=middle src="images/speedsnd.gif">.) The smaller the soliton's width (<b>~1/<i>k</i></b>), the faster it propagates. <p> The lattice is compressed by the value <b>2<i>k</i>/<i>b</i></b> around a soliton (we assume <nobr><b><i>b</i> > 0</b></nobr>). Due to such a compression, we can speak of the soliton's <nobr>"<b>mass</b>" = <b>2<i>mk</i>/<i>b</i></b>.</nobr> <p> A more complete description of the Toda solitons you can find in the book: <br> M. Toda, <i>Theory of Nonlinear Lattices</i>, (Springer-Verlag, New York 1989). <hr width=75%> The above <font color=red><b><i>applet</i></b></font> allows you to watch propagating solitons in the lattice of 15 particles with fixed boundaries. The evolution of the chain is calculated by the molecular-dynamics technique. The parameters of the lattice are the following: <nobr><b><i>a</i>=1, <i>b</i>=1, <i>m</i>=1</b></nobr>. <p> You can launch either a <i>wide soliton</i> (<b><i>k</i>=0.5</b>) or a <i>sharp</i> one (<b><i>k</i>=1</b>). You can also see non-destructive collisions of <i>two sharp solitons</i>. <p> The time is shown in units of the shortest period of small amplitude plane wave vibrations, <img align=middle src="images/tmeql.gif">. Energy is shown in arbitrary units. The <font color=red><b>kinetic</b></font> energy of the particle and the <font color="ccaa00"><b>potential</b></font> energy of the bond are shown as the <font color=red><b>red</b></font> and the <font color="#ccaa00"><b>yellow</b></font> bars, respectively. <p> If you wait for a while you will see a power spectrum of the particles' vibrations. It will be shown in the left panel. As the time of the evolution goes the spectrum resolution improves. The frequency unit is the maximal plane wave frequency, <img align=middle src="images/wmeql.gif">. <hr> <font size=-2> Last modified: December 1, 1996<br> <A HREF="http://www.lightlink.com/sergey/">Sergey Kiselev</a>, <i><a href="mailto:sergey@lightlink.com">sergey@lightlink.com</a></i><br> </font> </BODY>