Ëîêàëèçîâàííûå âîçáóæäåíèÿ, ðàñïðîñòðàíÿþùèåñÿ â ñèñòåìå ñ ïîñòîÿííîé ñêîðîñòüþ è âçàèìîäåéñòâóþùèå äðóã ñ äðóãîì áåç èçìåíåíèÿ èõ ôîðìû íàçûâàþò cîëèòîíàìè.  òå÷åíèå ñòîëêíîâåíèÿ cîëèòîííîå ðåøåíèå íå ìîæåò áûòü ïðåäñòàâëåíî êàê ëèíåéíàÿ êîìáèíàöèÿ ñîëèòîííûõ ðåøåíèé. Ïîñëå ñòîëêíîâåíèÿ ñîëèòîíû âîññòàíàâëèâàþò ôîðìó. Åäèíñòâåííûé ðåçóëüòàò ñòîëêíîâåíèÿ - ôàçîâûé ñäâèã.

Íà ýòèõ ñòðàíèöàõ ìû ïðåäñòàâëÿåì ñîëèòîíû óðàâíåíèÿ sine-Gordon è èõ ñòîëêíîâåíèÿ.

Óðàâíåíèå sine-Gordon

èãðàåò âàæíóþ ðîëü â ìíîãèõ îáëàñòÿõ ôèçèêè. Ýòî îäíà èç ïðîñòåéøèõ ìîäåëåé åäèíîé òåîðèè ïîëÿ, ïðèìåíÿåòñÿ â òåîðèè äèñëîêàöèé â ìåòàëëàõ, â òåîðèè äæîçåôñîíîâñêèõ ïåðåõîäîâ è ò.ä. Îíî ìîæåò èñïîëüçîâàòüñÿ òàêæå â ìîäåëèðîâàíèè íåêîòîðûõ áèîëîãè÷åñêèõ ïðîöåññîâ, â ÷àñòíîñòè äèíàìèêè ÄÍÊ.

Ìíîãî-ñîëèòîííûå ðåøåíèÿ: íà ýòèõ ñòðàíèöàõ ìû äåìîíñòðèðóåì íàõîæäåíèå ìíîãî-ñîëèòîííûõ ðåøåíèé óðàâíåíèÿ ñèíóñ-Ãîðäîí ñ èñïîëüçîâàíèåì ïðåîáðàçîâàíèÿ Áýêëóíäà. Àíèìàöèîííîå ïðåäñòàâëåíèå è 3-ìåðíàÿ ãðàôèêà ïîçâîëÿþò âèçóàëèçèðîâàòü îäèíî- (Êèíê, Àíòèêèíê), äâóõ- (Êèíê-Êèíê, Êèíê-Aíòèêèíê, Áðèçåð), òðåõ- ñîëèòîííûå ðåøåíèÿ è îñíîâíûå ñâîéñòâà ñòîëêíîâåíèé ñîëèòîíîâ.

Ìîäåëü öåïî÷êè ìàÿòíèêîâ: ñîëèòîííûå ðåøåíèÿ óðàâíåíèÿ ñèíóñ-Ãîðäîí è ñòîëêíîâåíèÿ ñîëèòîíîâ âèçóàëèçèðóþòñÿ ñ èñïîëüçîâàíèåì öåïî÷êè ñîåäèíåííûõ ìàÿòíèêîâ, äèñêðåòíàÿ ñèñòåìà óðàâíåíèé äëÿ êîòîðîé èìååò óðàâíåíèå ñèíóñ-Ãîðäîí êàê êîíòèíóàëüíûé àíàëîã.

Ìîäåëü óïðóãîé ëåíòû: ìíîãî-ñîëèòîííûå ðåøåíèÿ óðàâíåíèÿ ñèíóñ-Ãîðäîíà ïðåäñòàâëåíû ñ èñïîëüçîâàíèåì ìîäåëè óïðóãîé ëåíòû, êîòîðàÿ ïîçâîëÿåò ïðåäñòàâèòü ñîëèòîíû êàê çàðÿæåííûå ÷àñòèöû. Äëÿ àíèìàöèîííîãî ïðåäñòàâëåíèÿ ìû ñòàðàëèñü âûáðàòü ïàðàìåòðû, ÷òîáû ïðîäåìîíñòðèðîâàòü çàêîí ñîõðàíåíèÿ çàðÿäà. 

Ýôôåêòû äèñêðåòíîñòè: ïðåäñòàâëåíû ýôôåêòû íåóïðóãèõ ñòîëêíîâåíèé â ñëàáî-äèñêðåòíîé ñèñòåìå ñèíóñ-Ãîðäîí. Ïîä÷åðêíóòû ñóùåñòâîâàíèå è çíà÷åíèå ñåïàðàòðèñíûõ ðåøåíèé è íåóñòîé÷èâûõ ñòîõàñòè÷åñêèõ ñëîåâ.

• Dynamical Chaos in Classical and Quantum Physics. Budker Institute of Nuclear Physics, Novosibirsk, Russia August 4-9, 2003. Dedicated to 75-th birthday of Professor Boris Chirikov.
• Symmetry in Nonlinear Mathematical Physics. Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv (Kiev), Ukraine, June 23-29, 2003.

Òåñòîâûå ïðèìåðû äëÿ Maple-ïàêåòà íàõîæäåíèÿ ìíîãî-ñîëèòîííûõ ðåøåíèé óðàâíåíèÿ ñèíóñ-Ãîðäîí ñ èñïîëüçîâàíèåì ïðåîáðàçîâàíèÿ Áýêëóíäà (À.Å. Ìèðîøíè÷åíêî, A.A. Âàñèëüåâ, Ñ.Â. Äìèòðèåâ).

• Solitons Home Page (Heriot-Watt University, Edinburgh)
• Soliton-Lab Art Gallery.  Many Faces of Solitons (Kyoto University)
• Nonlinear Physics Group (Australian National University)
• Klaus Brauer's SOLITON Page (University of Osnabrück)
• Solitons.  Sine-Gordon pulson (Technical University of Denmark)
• Light Bullet Home Page (Simon Fraser University)
• Harmonic maps and integrable systems (ed. A.P. Fordy and J.C. Wood)

Ìíîãî-ïîëåâûå ñîëèòîíû: íà ýòèõ ñòðàíèöàõ ìû ïðåäñòàâëÿåì ìîäåëü, êîòîðàÿ ïîääåðæèâàåò óñòîé÷èâûå ïåðèîäè÷åñêèå ñòðóêòóðû è ìíîãî-ïîëåâûå ñîëèòîíû. Îáû÷íî ñîëèòîíû îïèñûâàþò îäíîé íåïðåðûâíîé ôóíêöèåé. N-ïîëåâûå ñîëèòîíû îïèñûâàþòñÿ ñ ïðèìåíåíèåì N ôóíêöèé. Äëÿ èõ íàõîæäåíèÿ ñòðîÿòñÿ ìíîãî-ïîëåâûå îáîáùåííûå êîíòèíóàëüíûå ìîäåëè. Çäåñü ïðåäñòàâëåíû íåêîòîðûå Internet ñàéòû ñ èíôîðìàöèåé î îáîáùåííûõ êîíòèíóàëüíûõ ìîäåëÿõ (Êîññåðà, ìèêðîïîëÿðíûõ, ãðàäèåíòíîãî òèïà, íåëîêàëüíûõ, äð.)
 

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Ðîòîáðèçåðû â ëåñòíèöå äæîçåôñîíîâñêèõ êîíòàêòîâ, èõ ïðåäñòàâëåíèå â ðàìêàõ ìåõàíè÷åñêîé àíàëîãèè ìàÿòíèêîâ äàíî íà ýòîé ñòðàíèöå.

• LOCNET: Localisation by Nonlinearity and Spatial Discreteness, and Energy Transfer, in Crystals, Biomolecules and Josephson Arrays
• Sergej Flach's Discrete Breather Homepage (Max-Planck-Institut für Physik komplexer Systeme, Dresden)
• Intrinsic Localized Modes (Cornell University)
• Breather solutions of the topological discrete sine-Gordon model (University of Texas)

• Óåäèíåííûå âîëíû (ñîëèòîíû). Â.À.Àëåøêåâè÷, Ë.Ã.Äåäåíêî, Â.À.Êàðàâàåâ (Ôèçè÷åñêèé ôàêóëüòåò ÌÃÓ)
• Å. Í. Ïåëèíîâñêèé "Ñîëèòîíû â âîäå".
• À.Â.Ñàâèí Íåáîëüøèå êîëëåêöèè ïîñëåäíèõ ðàáîò.
• Î.Ì. Êèñåëåâ  Ãàëåðåÿ: ïîðòðåòû èçâåñòíûõ ñîëèòîíîâ.
• Â.Þ. Íîâîêøåíîâ "Ìàòåìàòè÷åñêèå ìîäåëè â åñòåñòâîçíàíèè. Ââåäåíèå â òåîðèþ ñîëèòîíîâ."