{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 1 16 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 1 16 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 1 16 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 1 16 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 1 16 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Co urier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Out put" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE " " -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Author" -1 19 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 8 8 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 3" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 10 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 260 1 {CSTYLE "" -1 -1 "Times" 1 10 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 261 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 262 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 4 "Sine" }{TEXT 256 39 "-Gor don Solitons and Soliton Collisions" }}{PARA 19 "" 0 "" {TEXT -1 0 "" }{TEXT 257 2 "by" }}{PARA 19 "" 0 "" {TEXT 270 60 "Andrey E. Miroshnic henko, andrey.miroshnichenko@anu.edu.au," }}{PARA 19 "" 0 "" {TEXT 269 55 " Aleksey A. Vasiliev, aleksey.vasiliev@tversu.ru," }} {PARA 257 "" 0 "" {TEXT -1 0 "" }{TEXT 271 47 "Sergey V. Dmitriev, ser gey@iis.u-tokyo.ac.jp " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 19 " " 0 "" {TEXT -1 192 "NOTE: This worksheet demonstrates the use of Mapl e for obtaining multi-soliton solutions to the sine-Gordon equation, 2 D-animation and 3D graphics for representation of the soliton collisio ns." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 0 "" }{TEXT 258 12 "Introduct ion" }}{PARA 0 "" 0 "" {TEXT -1 376 "The localized excitations propaga ting in a system with constant velocity and colliding with each other \+ without change in their shapes are called solitons. During the collisi on of solitons the solution cannot be represented as a linear combinat ion of two soliton solutions but after the collision solitons recover \+ their shapes and the only result of collision is a phase shift." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "In the wo rksheet we present the sine-Gordon solitons and their collisions. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "The sine- Gordon equation " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "diff(phi(x,t),t$2)-diff(phi(x,t ),x$2)+sin(phi(x,t))=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(-%%diffG 6$-%$phiG6$%\"xG%\"tG-%\"$G6$F,\"\"#\"\"\"-F&6$F(-F.6$F+F0!\"\"-%$sinG 6#F(F1\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 306 "plays an important role \+ in many branches of physics. It provides one of the simplest models of the unified field theory, can be found in the theory of dislocations \+ in metals, in the theory of Josephson junctions and so on. It can be u sed also in interpreting certain biological processes like DNA dynamic s." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 259 63 "Backlund T ransformations and Nonlinear Superposition Principle " }}{PARA 0 "" 0 "" {TEXT -1 333 "In the last century a Sweden mathematician Backlund, \+ considering the geometry of surfaces with constant negative curvature, showed a way to obtain the hierarchy of sine-Gordon solutions when a \+ new solution can be build on the bases of known solutions. The transfo rmation, as applied to the sine-Gordon equation, has the form Ref. [1] " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "Backl_Trans:=\{diff((ph i(xi,tau)+psi(xi,tau))/2,xi)=a*sin((phi(xi,tau)-psi(xi,tau))/2), diff( (phi(xi,tau)-psi(xi,tau))/2,tau)=(1/a)*sin((phi(xi,tau)+psi(xi,tau))/2 )\};" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%,Backl_TransG<$/,&-%%diffG6$ -%$phiG6$%#xiG%$tauGF.#\"\"\"\"\"#*&F0F1-F)6$-%$psiGF-F.F1F1*&%\"aGF1- %$sinG6#,&F+F0*&#F1F2F1F6F1!\"\"F1/,&-F)6$F+F/F0*&#F1F2F1-F)6$F6F/F1F@ *&-F;6#,&F+F0*&F0F1F6F1F1F1F9F@" }}}{PARA 0 "" 0 "" {TEXT -1 6 "where \+ " }{XPPEDIT 18 0 "xi = (x-t)/2,tau = (x+t)/2;" "6$/%#xiG*&,&%\"xG\"\" \"%\"tG!\"\"F(\"\"#F*/%$tauG*&,&F'F(F)F(F(F+F*" }{TEXT -1 3 " , " } {XPPEDIT 18 0 "a;" "6#%\"aG" }{TEXT -1 28 "- transformation parameter, " }{XPPEDIT 18 0 "phi,psi;" "6$%$phiG%$psiG" }{TEXT -1 30 " - solutio ns of the equation " }{XPPEDIT 18 0 "diff(phi(xi,tau),xi,tau) = sin(p hi(xi,tau));" "6#/-%%diffG6%-%$phiG6$%#xiG%$tauGF*F+-%$sinG6#-F(6$F*F+ " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "Assuming that the diagram for the construction of the sol utions " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }{OLE 1 41121 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N::wyyyyyB:jy yyyyI::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::uyyyyyaJ:FB^;f:n:v:>;F;N;V;nYf;n;v;;JBB:]:_:e:c:g:k:i:m:q:o:s:w: u:y:=;;[:N>^>V>f>v>n>>?N?F?>ANB^?f?n?v?>@:y;;\\:nYnyyyYA^:fBWMtNHm=;:::::::n:;`:Z@[::JRTLEHv@Io\\ Aj;JZ]<<:=ja^GE=;:::::::::N;?R:yyyyyy:>:<::::::JDJ:J\\FFcmj>:;:::::::: :N;:AJ:vYxY:B::::::^:>ZJJ:jR^D=Mk^GJ:::::::::n;?R:ryC:yayA:<:::::: =:;f=:G:IB:>;;j>J?>:Q:SJ:f;;JAjAJBB:]:_:a:c:e:gJ:v<>=F=N =V=^=f=n=v=>^:B:=C:N>V>^>f>n>v>>?F?N?V?^?f?n?v?>@:_;a;c;e;g;i;k;m; 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Z::::::JywYB:::::::::::::yay=J:B:::::::::::::::::::jysy:>:<::::::::::: :dL;;B:::::bD:<:::::::::::6:" }{TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "is commutative, one ca n eliminate the partial derivatives and find the analytical expression of the Backlund transformations" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "phi[n+1]:=phi[n-1]+4*arctan(((a[1]+a[2])/(a[1]-a[2]))*(tan((ph i1[n]-phi2[n])/4)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$phiG6#,&% \"nG\"\"\"F)F),&&F%6#,&F(F)F)!\"\"F)*&\"\"%F)-%'arctanG6#*&*&,&&%\"aG6 #F)F)&F86#\"\"#F)F)-%$tanG6#,&&%%phi1G6#F(#F)F0*&#F)F0F)&%%phi2GFCF)F. F)F),&F7F)F:F.F.F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 6 "where " } {XPPEDIT 18 0 "a[1];" "6#&%\"aG6#\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[2];" "6#&%\"aG6#\"\"#" }{TEXT -1 41 " - are the parameters of \+ transformation, " }{XPPEDIT 18 0 "phi[n];" "6#&%$phiG6#%\"nG" }{TEXT -1 30 " is the n-parametric solution." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 62 "This formula gives a way to build the multi-soliton solutions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 82 "Procedure \"Backlund\" realizes the algorithm of B acklund transformation in Maple V." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "Backlund := proc (a1,a2,phi0,phi1,phi2) RETURN(phi0+ 4*arctan(((a1+a2)*tan((phi1-phi2)/4))/(a1-a2))) end;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)BacklundGf*6'%#a1G%#a2G%%phi0G%%phi1G%%phi2G6\"F, F,-%'RETURNG6#,&9&\"\"\"*&\"\"%F2-%'arctanG6#*&*&,&9$F29%F2F2-%$tanG6# ,&9'#F2F4*&#F2F4F29(F2!\"\"F2F2,&F;F2F " 0 "" {MPLTEXT 1 0 117 "Soliton := proc (x,t,a,x_0)\nlocal xi,tau:\nxi:=(x-x_0+t)/2; \nta u:=(x-x_0-t)/2; \nRETURN(4*arctan(exp(a*xi+tau/a))) end;" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#>%(SolitonGf*6&%\"xG%\"tG%\"aG%$x_0G6$%#xiG%$tau G6\"F.C%>8$,(9$#\"\"\"\"\"#*&#F5F6F59'F5!\"\"*&F4F59%F5F5>8%,(F3F4*&#F 5F6F5F9F5F:*&#F5F6F5FF5FPF:F5\"\"%F.F.F." }}}{PARA 0 "" 0 "" {TEXT -1 6 "where " } {XPPEDIT 18 0 "x,t;" "6$%\"xG%\"tG" }{TEXT -1 31 " - space and time co ordinates,\n" }{XPPEDIT 18 0 "a;" "6#%\"aG" }{TEXT -1 52 " - parameter of transformation for Kink (Antikink),\n" }{XPPEDIT 18 0 "x_0;" "6#%$ x_0G" }{TEXT -1 40 " - initial position of Kink (Antikink).\n" }} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 4 "Kink" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with(plottools):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name cha ngecoords has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "v[K]:=0.1: # velocity of the Kink" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "x_K:=-4*sqrt(1-v[K]^2): # initial state of the Kink" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "a:=sqrt((1-v[K])/(1 +v[K])): # parameter of transformation" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "L:=40*sqrt(1-v[K]^2): T:=(L-2*x_K)/v[K]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "K:=(x,t)->Soliton(x,t,a,x_K):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "animate(K(x,t),x=0..L,t=0..T,title=\"Kink\",color=red ,view=[0..L,-1..2*Pi+1]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} {PARA 13 "" 1 "" {GLPLOT2D 522 167 167 {PLOTDATA 2 "6&-%(ANIMATEG627#- %'CURVESG6$7S7$$\"\"!F-$\"3!*f#fA%4$*4i!#<7$$\"+v>c\"H)!#5$\"3UhI@/iM^ iF07$$\"+&R7$e;!\"*$\"3Q\"3CI&yMpiF07$$\"+#fou[#F:$\"3A0K$\\brrF'F07$$ \"+!zCmJ$F:$\"3%e')G'R=$G'F07$$\"+dI !z2\"!\")$\"3G9glhQ=$G'F07$$\"+w'=3;\"F[p$\"3yJw&*yY=$G'F07$$\"+'HMPC \"F[p$\"3[]a:M]=$G'F07$$\"+;*\\mK\"F[p$\"3]&yB&)=&=$G'F07$$\"+Obc49F[p $\"3/))>hb_=$G'F07$$\"+b6[#\\\"F[p$\"3W+%oZG&=$G'F07$$\"+vnRv:F[p$\"3) [sRuH&=$G'F07$$F9F[p$\"3Agm%HI&=$G'F07$$\"+:!G7u\"F[p$\"3Wk*R`I&=$G'F0 7$$\"+NO9C=F[p$\"3w*3!Q1`=$G'F07$$\"+a#fq!>F[p$\"3SE@$oI&=$G'F07$$\"+u [(**)>F[p$\"3!3eGqI&=$G'F07$$\"+%\\!*G2#F[p$\"3ifR62`=$G'F07$$\"+7h!e: #F[p$\"3/l5:2`=$G'F07$$\"+LnrI&=$G'F07$$\"+_tj@BF[p$\"3W *>urI&=$G'F07$$\"+tHb/CF[p$\"3+Xs<2`=$G'F07$$F?F[p$\"3Ep&yrI&=$G'F07$$ \"+7UQqDF[p$\"3#R9zrI&=$G'F07$$\"+J)*H`EF[p$\"3Q%RzrI&=$G'F07$$\"+_a@O FF[p$\"3u-&zrI&=$G'F07$$\"+r58>GF[p$\"3#)\\&zrI&=$G'F07$$\"+#pY?!HF[p$ \"39r&zrI&=$G'F07$$\"+5B'\\)HF[p$\"3-!ezrI&=$G'F07$$\"+Jz(y1$F[p$\"3c$ ezrI&=$G'F07$$\"+]Nz]JF[p$\"3M&ezrI&=$G'F07$$\"+r\"4PB$F[p$\"3C'ezrI&= $G'F07$$FDF[pF\\x7$$\"+5/a*R$F[pF\\x7$$\"+HgX#[$F[pF\\x7$$\"+];PlNF[pF \\x7$$\"+psG[OF[pF\\x7$$\"+!*G?JPF[pF\\x7$$\"+3&=T\"QF[pF\\x7$$\"+HT.( *QF[pF\\x7$$\"+[(\\*zRF[pF\\x-%'COLOURG6&%$RGBG$\"*++++\"F[pF,F,7#-F(6 $7S7$F,$\"3_Vk*3H-Sf%F07$F2$\"3i*4Z#oEy6bF07$F8$\"3Ow^$*)oFX%fF07$F>$ \"3iT)e;[>d8'F07$FC$\"3>%p\\MOt!>iF07$FH$\"3_^w#eV?`D'F07$FM$\"3+q16D^ 2riF07$FR$\"3iWGqIA#zF'F07$FW$\"3/)>+D)z*3G'F07$Ffn$\"3]iM))R7>#G'F07$ F[o$\"3_4iI(G`FG'F07$F_o$\"3/Gf6_v*HG'F07$Fdo$\"3))3_;4P5$G'F07$Fio$\" 3a0(G[%)\\JG'F07$F_p$\"39>aI&*)pJG'F07$Fdp$\"3#>o1#4'yJG'F07$Fip$\"3eg DD'R#=$G'F07$F^q$\"3_:f4US=$G'F07$Fcq$\"3'*)[wtv%=$G'F07$Fhq$\"3?XhBo] =$G'F07$F]r$\"3O&>NL?&=$G'F07$Far$\"3')***[?E&=$G'F07$Ffr$\"3m9fc(G&=$ G'F07$F[s$\"3W>bl)H&=$G'F07$F`s$\"3+U]Z.`=$G'F07$Fes$\"3a)fpbI&=$G'F07 $Fjs$\"3;)))zkI&=$G'F07$F_t$\"3#))\\voI&=$G'F07$Fdt$\"3iIu/2`=$G'F07$F it$\"3=_@72`=$G'F07$F^u$\"3wDY:2`=$G'F07$Fbu$\"3-Q(orI&=$G'F07$Fgu$\"3 yr[<2`=$G'F07$F\\v$\"3?Pv<2`=$G'F07$Fav$\"3G'pyrI&=$G'F07$Ffv$\"3))*>z rI&=$G'F07$F[w$\"3O=%zrI&=$G'F07$F`w$\"3S8&zrI&=$G'F07$Few$\"39b&zrI&= $G'F07$Fjw$\"3#HdzrI&=$G'F07$F_x$\"3!4ezrI&=$G'F07$FaxFbw7$FdxFgwFfxFi xF\\yF_yFbyFeyFhy7#-F(6$7S7$F,$\"3?r^O\"[7)=O!#=7$F2$\"3\\$R+$euRJ#)Ff bl7$F8$\"3YC\")o!Hs5z\"F07$F>$\"3:\"e)fF,ETLF07$FC$\"3izJ:G=_%y%F07$FH $\"3G'fSBl>hg&F07$FM$\"31M&)o-bj')fF07$FR$\"3Y/<[lM6ahF07$FW$\"3b,Xe>^ 2FiF07$Ffn$\"3o$pMFi)zeiF07$F[o$\"3oYNH\"z'esiF07$F_o$\"3)G3=C?z&yiF07 $Fdo$\"3%Qu$e,N=\"G'F07$Fio$\"3-yY*fK:BG'F07$F_p$\"3')R-\"\\@2GG'F07$F dp$\"3=Ji&*))4-$G'F07$Fip$\"3OOSx%*Q6$G'F07$F^q$\"3KdgZrU:$G'F07$Fcq$ \"3[G%=\">=<$G'F07$Fhq$\"3#)oBHX%zJG'F07$F]r$\"3]-RhfF=$G'F07$Far$\"35 /@,+U=$G'F07$Ffr$\"3o\"e1g#[=$G'F07$F[s$\"3ogE1)4&=$G'F07$F`s$\"3_XxH; 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0 "" {MPLTEXT 1 0 252 "TwoSoliton : = proc (x,t,a1,x_1,a2,x_2)\nlocal phi0,phi1,phi2:\nphi0[0]:=(x,t)->0; \nphi1[1]:=(x,t)->Soliton(x,t,a1,x_1);\nphi2[1]:=(x,t)->Soliton(x,t,a2 ,x_2);\nphi1[2]:=(x,t)->Backlund(a1,a2,phi0[0](x,t),phi1[1](x,t),phi2[ 1](x,t));\nRETURN(phi1[2](x,t)) end;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%+TwoSolitonGf*6(%\"xG%\"tG%#a1G%$x_1G%#a2G%$x_2G6%%%phi0G%%phi1G% %phi2G6\"F1C'>&8$6#\"\"!F7>&8%6#\"\"\"f*6$F'F(F16$%)operatorG%&arrowGF 1-%(SolitonG6&9$9%T$T&F1F16&F)9&F*9'>&8&F;f*F>F1F?F1FBF1F16&F+9(F,9)>& F:6#\"\"#f*F>F1F?F1-%)BacklundG6'FGFH-&T'F66$FEFF-&T)F;Fhn-&T+F;FhnF1F 16,F)FJF+FQF.F5F/F:F0FN-%'RETURNG6#-FTFhnF1F1F1" }}}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "x,t;" "6$%\"xG%\"tG" }{TEXT -1 31 " - space and time coordinates,\n" }{XPPEDIT 18 0 "a1,a2;" "6$%#a1G%#a 2G" }{TEXT -1 33 " - parameters of transformation,\n" }{XPPEDIT 18 0 " x_1,x_2;" "6$%$x_1G%$x_2G" }{TEXT -1 22 " - initial positions.\n" }} {PARA 0 "" 0 "" {TEXT -1 62 "Examples of two-soliton solutions and two -soliton collisions: " }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 19 "Kink-Kin k Collision" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "v[K_K]:=0.2: \+ # velocity of the Kink-Kink" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "x _K_K:=0: # initial state of the Kink-Kink " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "a[1]:=-sqrt((1-v[K_K])/(1+v[K_K])): # parameters of \+ " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "a[2]:=sqrt((1+v[K_K])/(1-v[K_K] )): # transformation " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "L:=10/sq rt(1-v[K_K]^2): T:=L/v[K_K]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "K_K :=(x,t)->TwoSoliton(x,t,a[1],x_K_K,a[2],x_K_K):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "animate(K_K(x,t),x=-3*L..3*L,t=-4*T..4*T,title=\"Kin k-Kink Collison\",color=red,view=[-3*L..3*L,-2*Pi-1..2*Pi+1]);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 520 173 173 {PLOTDATA 2 "6&-%(ANIMATEG627#-%'CURVESG6$7S7$$!+y@'=1$!\")$!3 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It seems that in the collision they repel each other, however, they actually pass through \+ each other without change in their velocities." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 23 "Kink-AntiKink Co llision" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "v[K_AK]:=0.2: # velocity of the Kink-AntiKink" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "x _K_AK:=0: # initial state of the Kink-AntiKink " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "a[1]:=sqrt((1-v[K_AK])/(1+v[K_AK])): # paramete rs of " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "a[2]:=sqrt((1+v[K_AK])/(1 -v[K_AK])): # transformation " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "L :=10/sqrt(1-v[K_AK]^2): T:=L/v[K_AK]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "K_AK:=(x,t)->TwoSoliton(x,t,a[1],x_K_AK,a[2],x_K_AK):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "animate(K_AK(x,t),x=-3*L..3*L,t=-2*T..2* T,title=\"Kink-AntiKink Collision\",color=red,view=[-3*L..3*L,-2*Pi-1. .2*Pi+1]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 13 "" 1 "" 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-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "which is realized in the following procedure" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 309 "ThreeSoliton := proc (x,t,a1,x_1,a 2,x_2,a3,x_3)\nlocal phi,phi1,phi2,phi3 :\nphi2[1]:=(x,t)->Soliton(x, t,a2,x_2);\nphi1[2]:=(x,t)->TwoSoliton(x,t,a1,x_1,a2,x_2);\nphi2[2]:=( x,t)->TwoSoliton(x,t,a2,x_2,a3,x_3);\nphi1[3]:=(x,t)->Backlund(a3,a1,p hi2[1](x,t),phi2[2](x,t),phi1[2](x,t));\nRETURN(Re(phi1[3](x,t))) end ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%-ThreeSolitonGf*6*%\"xG%\"tG%#a 1G%$x_1G%#a2G%$x_2G%#a3G%$x_3G6&%$phiG%%phi1G%%phi2G%%phi3G6\"F4C'>&8& 6#\"\"\"f*6$F'F(F46$%)operatorG%&arrowGF4-%(SolitonG6&9$9%T$T&F4F46&F+ 9(F,9)>&8%6#\"\"#f*F&F8FMf*F&FL6#\"\"$f*F " 0 "" {MPLTEXT 1 0 37 "v[K]:= 0: # velocity of the Kink" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 " v[B]:=0.5: # velocity of the Breather" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "omega:=0.154: # frequency of the Breather" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "R:=sqrt((1-v[B])/(1+v[B])): " }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 21 "eta:=sqrt(1-omega^2):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "a[1]:=R*(eta+I*omega): # parameters of " }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "a[2]:=sqrt((1-v[K])/(1+v[K])): # \+ transformation " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "a[3]:=R*(eta-I*o mega):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "T:=evalf(2*Pi/omega): " } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "L:=T*abs(v[B])/(sqrt(1-v[B]^2)):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "x_K:=L/2: # initial state of the Kink " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "x_B:=0: # init ial state of the Breather" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "K_B:=( x,t)->Re(ThreeSoliton(x,t,a[1],x_B,a[2],x_K,a[3],x_B)):" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "2D animation of th e Kink-Breather interaction. 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The shift can be found from the formula " }{XPPEDIT 18 0 "Delt a[K] = -4*arctanh*sqrt((1-omega^2)*(1-v[B]^2));" "6#/&%&DeltaG6#%\"KG, $*(\"\"%\"\"\"%(arctanhGF+-%%sqrtG6#*&,&F+F+*$%&omegaG\"\"#!\"\"F+,&F+ F+*$&%\"vG6#%\"BGF4F5F+F+F5" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT 256 59 "Standing Kink and Moving Small Amplitude Breather collis ion" }}{PARA 0 "" 0 "" {TEXT -1 113 "We input parameters and, with the use of the procedure, build the Kink-Breather solution with the Kink \+ standing (" }{XPPEDIT 18 0 "v[K] = 0;" "6#/&%\"vG6#%\"KG\"\"!" }{TEXT -1 5 ") at " }{XPPEDIT 18 0 "x = x_K;" "6#/%\"xG%$x_KG" }{TEXT -1 40 " and Breather moving with the frequency " }{XPPEDIT 18 0 "omega;" "6#% &omegaG" }{TEXT -1 14 " and velocity " }{XPPEDIT 18 0 "v[B];" "6#&%\"v G6#%\"BG" }{TEXT -1 2 " ." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "v[K]:=0: # velocity of the Kink" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "v[B]:=0.85: # velocity of the Breather" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 41 "omega:=0.99: # frequency of the Breather" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "R:=sqrt((1-v[B])/(1+v[B])): " }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 21 "eta:=sqrt(1-omega^2):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "a[1]:=R*(eta+I*omega): # parameters of " }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "a[2]:=sqrt((1-v[K])/(1+v[K])): # \+ transformation " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "a[3]:=R*(eta-I*o mega):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "T:=4*2*Pi/omega: " }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "L:=0.75*T*abs(v[B])/(sqrt(1-v[B]^2) ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "x_K:=L/2: # initial stat e of the Kink " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "x_B:=0: # \+ initial state of the Breather" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "K_ B:=(x,t)->ThreeSoliton(x,t,a[1],x_B,a[2],x_K,a[3],x_B):" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "2D animation of th e Kink-Breather interaction. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "ddx:=-2*arctanh(sqrt((1-omega^2)*(1-v[B]^2))):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "L0:=line([x_K-ddx,-4], [x_K-ddx,10], color=blue, linestyle=3):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "L1:=line([x_K+ddx ,-4], [x_K+ddx,10], color=black, linestyle=3):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "an_K_B:=animate(K_B(x,t),x=-0.5*L..1.5*L,t=-T..2*T,t itle=\"Standing Kink and Moving Small Amplitude Breather collision\",c olor=red,numpoints=260):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "display ([an_K_B,L0,L1]);" }}{PARA 13 "" 1 "" {GLPLOT2D 456 206 206 {PLOTDATA 2 "6&-%(ANIMATEG627%-%'CURVESG6$7`[l7$$!+3P6O:!\")$!+@=*Q9\"!#57$$!+W* *Q7:F.$!+)obra\"F17$$!+!=m')[\"F.$!+NQ>#p\"F17$$!+fFK7$$!+M)*fA 8F.$\"+32bf()FK7$$!+qg())H\"F.$\"+`Md15F17$$!+1B:v7F.$\"+#HK!R)*FK7$$! +U&G9D\"F.$\"+!Q,1H)FK7$$!+yZqF7F.$\"+%3@\")y&FK7$$!+95)R?\"F.$\"+HFB( y#FK7$$!+^sD!=\"F.$!+G]U8C!#77$$!+([Ll:\"F.$!+$=s&yGFK7$$!+B(4G8\"F.$! +\"H@*3[FK7$$!+ff346F.$!+v;]^eFK7$$!+&>i`3\"F.$!+1)f(ofFK7$$!+K%Q;1\"F .$!+IzVa_FK7$$!+oY\"z.\"F.$!+NR50RFK7$$!+/4>95F.$!+4W1\"=#FK7$$!++9n/* *!\"*$!+zaXJOFhp7$$!+jPVn'*F_s$\"+9%)H)G\"FK7$$!+Ah>I%*F_s$\"+vzsmDFK7 $$!+&[eH>*F_s$\"+_fOTLFK7$$!+[3sb*)F_s$\"+q%ygc$FK7$$!+6K[=()F_s$\"+6_ dvKFK7$$!+qbC\"[)F_s$\"+%p35d#FK7$$!+Lz+W#)F_s$\"+>H@)f\"FK7$$!+'Hqn+) F_s$\"+317C_Fhp7$$!+bE`pxF_s$!+!=Ft'\\Fhp7$$!+=]HKvF_s$!+eGNE8FK7$$!+ \"Qd]H(F_s$!+jSRv=FK7$$!+W(>y0(F_s$!+'*R5-@FK7$$!+.@e?oF_s$!+dX'Q,#FK7 $$!+mWM$e'F_s$!+;kJg;FK7$$!+Ho5YjF_s$!+`7_@6FK7$$!+)=p)3hF_s$!+KdqL\\F hp7$$!+^:jreF_s$\"+xRPu7Fhp7$$!+9RRMcF_s$\"+>-VnlFhp7$$!+ti:(R&F_s$\"+ %GgH.\"FK7$$!*k=*f^F.$\"+M2#GA\"FK7$$!*+\"oA\\F.$\"+\\0bA7FK7$$!*OVao% F.$\"+))y'[0\"FK7$$!*s0#[WF.$\"+cP5FwFhp7$$!*3o4@%F.$\"+jsU4SFhp7$$!*X IP(RF.$\"+GG&>u#!#87$$!*\"G\\OPF.$!+P?'H0$Fhp7$$!*A[@Fhp7$$!)BGzA/@Fhp7$$\"*BNF[\"F.$!+c]>w9Fhp7$$\"*(G(*>@?(F_[l7$$\"*^ 5s&>F.$\"++!e2^%!#97$$\"*:[W>#F.$\"+#*HqSrF_[l7$$\"*y&oJCF.$\"+xU&o?\" Fhp7$$\"*VB*oEF.$\"+g^Px9Fhp7$$\"*1hh!HF.$\"+C4f::Fhp7$$\"*q)RVJF.$\"+ (*egW8Fhp7$$\"*MO1Q$F.$\"+f\")e85Fhp7$$\"*)R(yh$F.$\"+smlzeF_[l7$$\"*h 6^&QF.$\"+a$oeQ\"F_[l7$$\"*E\\B4%F.$!+5g5'*p F_[l7$$\"*kDZY'F.$\"+CU\"3R*F_[l7$$\"*Gj>q'F.$\"+/)RX7\"Fhp7$$\"*\"4?R pF.$\"+Smlk7Fhp7$$\"*bQk<(F.$\"+wOT(F.$\"+lL(R]\"Fhp7 $$\"*$Q\"4l(F.$\"+i#*3t;Fhp7$$\"*Y^\"))yF.$\"+'oZ2$>Fhp7$$\"*6*QD\")F. $\"+lob@BFhp7$$\"*uEEO)F.$\"+cvy!*GFhp7$$\"*Qk)*f)F.$\"+E[d%o$Fhp7$$\" *--r$))F.$\"+n:s^ZFhp7$$\"*mRV2*F.$\"+wn6ZhFhp7$$\"*Hx:J*F.$\"+'3+n$zF hp7$$\"*%\\\")[&*F.$\"+SoQ?5FK7$$\"*d_gy*F.$\"+w(>dI\"FK7$$\"+@!HB+\"F .$\"+B([Rm\"FK7$$\"+&y_g-\"F.$\"+$pBQ6#FK7$$\"+\\lx\\5F.$\"+')zyzEFK7$ $\"+7.]t5F.$\"+.DV$R$FK7$$\"+wSA(4\"F.$\"+]@E&H%FK7$$\"+Sy%47\"F.$\"+Q V)oV&FK7$$\"+/;nW6F.$\"+r(fP)oFK7$$\"+o`Ro6F.$\"+pso=()FK7$$\"+J\">@> \"F.$\"+QCi/6F17$$\"+&*G%e@\"F.$\"+\">F)*R\"F17$$\"+fmcR7F.$\"+@n5uW3OF17$$\"+9Q\" F.$\"+7>3#H(F17$$\"+1Ij09F.$\"+32o$=*F17$$\"+qnNH9F.$\"+M'*>_6F_s7$$\" +M03`9F.$\"+O(\\sV\"F_s7$$\"+(H/oZ\"F.$\"+izkx;F.$\"+:Nu4 WF_s7$$\"+W1(Gk\"F.$\"+ISSkZF_s7$$\"+3Wfm;F.$\"+27Nj]F_s7$$\"+s\"=.p\" F.$\"+\"4$p4`F_s7$$\"+O>/9XQc=F.$\"+__i%4'F_s7$$ \"+#G3,)=F.$\"+a1SMhF_s7$$\"+Y?$Q!>F.$\"+I^zlhF_s7$$\"+5ebF>F.$\"+-#p0 >'F_s7$$\"+u&z7&>F.$\"+jp65iF_s7$$\"+QL+v>F.$\"+;0aDiF_s7$$\"+-rs)*>F. $\"+#>5xB'F_s7$$\"+l3XA?F.$\"+xFJZiF_s7$$\"+HY[0G'F_s7$$\"+Jg82BF.$\"+% H-6G'F_s7$$\"+%zf3L#F.$\"+7%R:G'F_s7$$\"+eNeaBF.$\"+EY)=G'F_s7$$\"+BtI yBF.$\"+Iw:#G'F_s7$$\"+(3J?S#F.$\"+9QP#G'F_s7$$\"+_[vDCF.$\"+2_a#G'F_s 7$$\"+7'y%\\CF.$\"+a6o#G'F_s7$$\"+xB?tCF.$\"+`*)y#G'F_s7$$\"+Th#p\\#F. $\"+*HuGG'F_s7$$\"+1*\\1_#F.$\"+r;%HG'F_s7$$\"+qOPWDF.$\"+IY*HG'F_s7$$ \"+Nu4oDF.$\"+Mg.$G'F_s7$$\"+&>@=f#F.$\"+:#oIG'F_s7$$\"+g\\a:EF.$\"+&3 $4$G'F_s7$$\"+C(o#REF.$\"+KA6$G'F_s7$$\"+*[#*Hm#F.$\"+np7$G'F_s7$$\"+` ir'o#F.$\"+f$QJG'F_s7$$\"+=+W5FF.$\"+os9$G'F_s7$$\"+yP;MFF.$\"+rV:$G'F _s7$$\"+Vv)yv#F.$\"+i,;$G'F_s7$$\"+28h\"y#F.$\"+-];$G'F_s7$$\"+s]L0GF. $\"+>\"pJG'F_s7$$\"+O)e!HGF.$\"+bE<$G'F_s7$$\"+,Ey_GF.$\"+*ovJG'F_s7$$ \"+hj]wGF.$\"+R#yJG'F_s7$$\"+E,B+HF.$\"+=.=$G'F_s7$$\"+!*Q&R#HF.$\"+Y> =$G'F_s7$$\"+bwnZHF.$\"+JJ=$G'F_s7$$\"+>9SrHF.$\"+@R=$G'F_s7$$\"+%=D^* HF.$\"+lV=$G'F_s7$$\"+X*[)=IF.$\"+SX=$G'F_s7$$\"+4FdUIF.$\"+GX=$G'F_s7 $$\"+tkHmIF.$\"+5W=$G'F_s7$$\"+Q--!4$F.$\"+bU=$G'F_s7$$\"+-Su8JF.$\"+K T=$G'F_s7$$\"+nxYPJF.$\"+!3%=$G'F_s7$$\"+G:>hJF.$\"+;T=$G'F_s7$$\"+#H: \\=$F.$\"+\\U=$G'F_s7$$\"+c!R'3KF.$\"+aW=$G'F_s7$$\"+@GOKKF.$\"+;Z=$G' F_s7$$\"+&e'3cKF.$\"+$*\\=$G'F_s7$$\"+Y.\")zKF.$\"+a_=$G'F_s7$$\"+6T`. 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The shift can be found from the formula " }{XPPEDIT 18 0 "Delta[K] = -4*arctanh*sqrt((1-omega^2) *(1-v[B]^2));" "6#/&%&DeltaG6#%\"KG,$*(\"\"%\"\"\"%(arctanhGF+-%%sqrtG 6#*&,&F+F+*$%&omegaG\"\"#!\"\"F+,&F+F+*$&%\"vG6#%\"BGF4F5F+F+F5" } {TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 258 " " 0 "" {TEXT -1 43 "Standing Breather and Moving Kink collision" }} {PARA 0 "" 0 "" {TEXT -1 56 "In this section we present the interactio n of standing (" }{XPPEDIT 18 0 "v[B] = 0;" "6#/&%\"vG6#%\"BG\"\"!" } {TEXT -1 7 ") at " }{XPPEDIT 18 0 "x = x_B;" "6#/%\"xG%$x_BG" } {TEXT -1 51 " Breather with the Kink moving with the velocity " } {XPPEDIT 18 0 "v[K];" "6#&%\"vG6#%\"KG" }{TEXT -1 2 " ." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "v[K]:=0.5: # velocity of the Kin k" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "v[B]:=0: # velocity of th e Breather" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "omega:=0.154: # frequ ency of the Breather" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "R:=sqrt((1- v[B])/(1+v[B])): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "eta:=sqrt(1-om ega^2):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "a[1]:=R*(eta+I*omega): \+ # parameters of " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "a[2] :=sqrt((1-v[K])/(1+v[K])): # transformation" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "a[3]:=R*(eta-I*omega):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "L:=10/sqrt(1-v[K]^2):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "x_ K:=-4*sqrt(1-v[K]^2): # initial state of the Kink" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 58 "x_B:=L/2: # initial state of the Bre ather " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "T:=(L-2*x_K)/v[K]:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "K_B:=(x,t)->ThreeSoliton(x,t,a[1],x _B,a[2],x_K,a[3],x_B):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "2D animation of the Kink-Breather interaction." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "ddx:=-arctanh(sqrt((1-omega^ 2)*(1-v[K]^2)))/sqrt(1-omega^2):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "L0:=line([x_B-ddx,-6], [x_B-ddx,12], color=blue, linestyle=3):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "L1:=line([x_B+ddx,-6], [x_B+ddx,12] , color=black, linestyle=3):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "an _B_K:=animate(K_B(x,t),x=-L..3*L,t=-T..2*T,title=\"Standing Breather M oving Kink collision\",color=red):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "display([an_B_K,L0,L1]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} {PARA 13 "" 1 "" {GLPLOT2D 459 247 247 {PLOTDATA 2 "6&-%(ANIMATEG627%- %'CURVESG6$7S7$$!+Q0qa6!\")$\"+'z$=$G'!\"*7$$!+$\\v%e5F.$\"+3d=$G'F17$ $!+$[/Di*F1$\"+ox=$G'F17$$!+MSDg')F1$\"+W@>$G'F17$$!+'e.!)p(F1$\"+_J?$ G'F17$$!+QJvNnF1$\"+O:B$G'F17$$!+\"p-Nx&F1$\"+_\\I$G'F17$$!+VAD6[F1$\" +7\\\\$G'F17$$!+$z,!\\QF1$\"+![')RG'F17$$!+X8v')GF1$\"++&e_G'F17$$!+(* 3]C>F1$\"+?,b)G'F17$$!)X]A'*F.$\"+7y1(H'F17$$\"\"!Feo$\"++$3\">jF17$$ \")X]A'*F.$\"+[(GhP'F17$$\"*!4]C>F.$\"+3;WBlF17$$\"*M^n)GF.$\"+))*f0!p F17$$\"*z,!\\QF.$\"+g*>@\"yF17$$\"*C_7\"[F.$\"+7W5%[*F17$$\"*p-Nx&F.$ \"+6da/6F.7$$\"*9`dt'F.$\"+Ki!3<\"F.7$$\"*f.!)p(F.$\"+\\h@f6F.7$$\"*.a -m)F.$\"+MM[k5F.7$$\"*Z/Di*F.$\"+G](R!*)F17$$\"+$\\v%e5F.$\"+;!oPW(F17 $$\"+P0qa6F.$\"+Gl/VnF17$$\"+$eD4D\"F.$\"+GdehkF17$$\"+H1:Z8F.$\"+'*p; 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5$\"+gH<$G'F5$\"+Ga9$G'F5$\"+O\"3IG'F5$\"+O\"*H#G'F5$\"+3YjyiF5$\"+cRr fiF5$\"+'p(4ihF5$\"+'zGJm&F5$\"+IOhoNF5$\"*SpIV*F5$\"+/%o7'=FJ$\"*sF$4 OFJ$\"*k[X*pFO$\"+K=Rb8FR$\"+K[ZCEFU$\"*g670&FU$\"*7M)e$*FZ$\"+[de(Q\" Fin-%+AXESLABELSG6%%\"xG%\"tGQ!6\"-%&TITLEG6#QHStanding~Breather~Movin g~Kink~collision6\"-%+PROJECTIONG6%$\"#q\"\"!$\"#XF]hw\"\"\"" 1 4 0 1 10 0 2 1 1 1 2 1.000000 3.000000 90.000000 1 0 "Curve 1" }}{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 90 "One can rotat e the 3D image in order to observe the process from different points o f view." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 222 "The presented graphics clearly show that the velocity and the osc illation frequency of the standing Breather, after the collision with \+ the moving Kink, are unchanged. The position of the Breather shifts af ter collision to " }{XPPEDIT 18 0 "X_B = x_B+Delta[B];" "6#/%$X_BG,&%$ x_BG\"\"\"&%&DeltaG6#%\"BGF'" }{TEXT -1 46 ". The shift can be found from the formula " }{XPPEDIT 18 0 "Delta[B] = 2*arctanh*sqrt((1-ome ga^2)*(1-v[K]^2))/sqrt(1-omega^2);" "6#/&%&DeltaG6#%\"BG**\"\"#\"\"\"% (arctanhGF*-%%sqrtG6#*&,&F*F**$%&omegaGF)!\"\"F*,&F*F**$&%\"vG6#%\"KGF )F3F*F*-F-6#,&F*F**$F2F)F3F3" }{TEXT -1 2 " ." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 10 "Conclusion" }} {PARA 0 "" 0 "" {TEXT -1 454 " In the present worksheet, we demonstrat ed the power of MAPLE deriving the multi-soliton solutions to the sine -Gordon equation with the use of Backlund transformations. We consider ed both real and complex parameters of transformations. The solutions \+ are given in the form of recursion procedures, which gives an efficien t representation of the \"soliton ladder\" idea. Animation and 3D grap hics help to visualize the basic properties of soliton collisions." }} {SECT 0 {PARA 260 "" 0 "" {TEXT 268 0 "" }{TEXT 267 10 "Disclaimer" } {TEXT -1 1 " " }}{PARA 259 "" 0 "" {TEXT -1 239 "While every effort ha s been made to validate the solutions in this worksheet, Waterloo Mapl e Inc. and the contributors are not responsible for any errors contain ed and are not liable for any damages resulting from the use of this m aterial." }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 10 "References" }} {PARA 0 "" 0 "" {TEXT -1 121 "[1] R.K.Dodd, J.C.Eilbeck, J.D.Gibbon, H .C.Morries, Solitons and Nonlinear Wave Equations (Academic Press, Lon don, 1982)." }}}}{MARK "0 5 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }