View source for Sine-Gordon equation

A relativistically-invariant equation in two space-time variables, of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085500/s0855001.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

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The name was suggested by M. Kruskal, in analogy with the linear [[Klein–Gordon equation|Klein–Gordon equation]] (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085500/s0855003.png" /> appears in place of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085500/s0855004.png" />). In characteristic (light-cone) variables the sine-Gordon equation has the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085500/s0855005.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

In both cases (1) and (2), the sine-Gordon equation admits a Lax representation

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085500/s0855007.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085500/s0855008.png" /> are linear operators and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085500/s0855009.png" />. This enables one to obtain solutions of the Cauchy problem by using the inverse scattering method.

The [[Cauchy problem|Cauchy problem]] for the sine-Gordon equation is formulated in the following way.

Case (1):

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Case (2):

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Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085500/s08550014.png" /> is the Schwartz space of rapidly-decreasing functions. Under certain additional restrictions on the initial conditions, the Cauchy problems (1) and (2) can be solved uniquely, and their solution sets coincide. The evolution of the scattering data for the corresponding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085500/s08550015.png" />-operators is given by explicit formulas, and solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085500/s08550016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085500/s08550017.png" /> can be found by using integral equations of Gel'fand–Levitan–Marchenko type.

The periodic problem for the sine-Gordon equation can be studied by means of an algebraic-geometric method (similar to the case of the [[Korteweg–de Vries equation|Korteweg–de Vries equation]]). In particular, one obtains explicit expressions for the finite-gap solutions of the sine-Gordon equation in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085500/s08550018.png" />-functions on the corresponding Abelian varieties.

The Hamiltonian version of the sine-Gordon equation (in case (1), for example) is the [[Hamiltonian system|Hamiltonian system]] with Hamiltonian

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and symplectic form

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This system is completely integrable, and replacing the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085500/s08550022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085500/s08550023.png" /> by the scattering data of the corresponding operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085500/s08550024.png" /> yields a canonical transformation into variables of action-angle type. The phase space is parametrized by canonically adjoint variables of three types:

1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085500/s08550025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085500/s08550026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085500/s08550027.png" />;

2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085500/s08550028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085500/s08550029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085500/s08550030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085500/s08550031.png" />;

3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085500/s08550032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085500/s08550033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085500/s08550034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085500/s08550035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085500/s08550036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085500/s08550037.png" />.

The total energy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085500/s08550038.png" /> and total momentum

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of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085500/s08550040.png" /> in terms of the new variables are as follows:

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In case (1) one also obtains a completely-integrable Hamiltonian system.

==An application to quantum field theory.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085500/s08550045.png" /> be a scalar field with [[Lagrangian|Lagrangian]]

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085500/s08550047.png" /> is the constant of interaction. The sine-Gordon equation is the [[Euler–Lagrange equation|Euler–Lagrange equation]] for this Lagrangian. In the quasi-classical quantization of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085500/s08550048.png" />, a fundamental role is played by the above formulas for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085500/s08550049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085500/s08550050.png" />. The first terms on their right-hand sides correspond to particles of mass <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085500/s08550051.png" /> and to particles of the ground field, respectively (cf. also [[Soliton|Soliton]]). The second and third terms correspond to localized solutions of the sine-Gordon equation, namely solitons and dual solitons with masses <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085500/s08550052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085500/s08550053.png" />, respectively. The system obeys the conservation law (of topological charge):

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Particles of the first and third types have charge 0, while those of the second type have charge <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085500/s08550055.png" />. Particles with equal charges repulse, while those with different charges attract. The presence of infinitely-many conservation laws means that the number of particles of each type is preserved under scattering; the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085500/s08550056.png" />-particle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085500/s08550057.png" />-matrix reduces to paired <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085500/s08550058.png" />-matrices (cf. [[Scattering matrix|Scattering matrix]]). Using integrals over trajectories (cf. [[Integral over trajectories|Integral over trajectories]]), one can compute the quantum corrections to the masses and to the quasi-classical <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085500/s08550059.png" />-matrix of solitons. One of the non-trivial properties of the above model is the appearance of a whole spectrum of particles (solitons), while the Lagrangian of the theory contains only one field. Furthermore, in the weak interaction approximation (that is, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085500/s08550060.png" /> is small), solitons are heavy particles and interact strongly.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M., et al. Ablowitz,   "Method for solving the sine-Gordon equation"  ''Phys. Rev. Letters'' , '''30'''  (1973)  pp. 1262–1264</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.A. Takhtadzhyan,   L.D. Faddeev,   "Essentially nonlinear one-dimensional model of classical field theory"  ''Teoret. Mat. Fiz.'' , '''21''' :  2  (1974)  pp. 160–174  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.A. [L.A. Takhtadzhyan] Tahtadžjan,   L.D. Faddeev,   "The Hamiltonian system connected with the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085500/s08550061.png" />"  ''Proc. Steklov Inst. Math.'' , '''142'''  (1979)  pp. 277–289  ''Trudy Mat. Inst. Akad. Nauk SSSR'' , '''142'''  (1976)  pp. 254–266</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.O. Kozel,   V.P. Kotlyarov,   "Almost-periodic solutions of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085500/s08550062.png" />"  ''Dokl. Akad. Nauk UkrSSR Ser. A'' , '''10'''  (1976)  pp. 878–881; 959  (In Russian)  (English abstract)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  V.E. Korepin,   L.D. Faddeev,   "Quantization of solitons"  ''Teoret. Mat. Fiz.'' , '''25''' :  2  (1975)  pp. 147–163  (In Russian)  (English abstract)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  L. Bianchi,   "Lezioni di geometria differenziale" , '''1–2''' , Zanichelli , Bologna  (1923–1927)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  S.P. Finikov,   "Deformation over a principal base and related problems in geometry" , Moscow-Leningrad  (1937)  (In Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  E.N. Pelinovskii,   "Some exact methods in nonlinear wave theory"  ''Izv. Vuzov. Radiofizika'' , '''19''' :  5–6  (1976)  pp. 883–901  (In Russian)</TD></TR></table>



====Comments====
There are  "canonical"  sine-Gordon type equations associated to every [[Kac–Moody algebra|Kac–Moody algebra]] [[#References|[a5]]], [[#References|[a4]]]. These are called Leznov–Saveliev systems. There are also Korteweg–de Vries type soliton equations associated to every Kac–Moody algebra, as well as modified Korteweg–de Vries type equations. These mKdV equations and Leznov–Saveliev systems associated to the same (extended) root system are  "in duality" , in the sense that the solutions of the one serve as symmetries of the other and vice versa, [[#References|[a4]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.J. Ablowitz,   H. Segur,   "Solitons and the inverse scattering transform" , SIAM  (1981)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L.D. Faddeev,   L.A. Takhtadzhyan,   "Hamiltonian methods in the theory of solitons" , Springer  (1967)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A.C. Newell,   "Solitons in mathematics and physics" , SIAM  (1985)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  V.G. Drinfel'd,   V.V. Sokolov,   "Lie algebras and equations of Korteweg–de Vries type"  ''J. Soviet Math.'' , '''30'''  (1985)  pp. 1975–2005  ''Itogi Nauk. i Tekhn. Sovrem. Probl. Mat.'' , '''24'''  (1984)  pp. 81–180</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  A.N. Leznov,   "On complete integrability of a nonlinear system of partial differential equations in two-dimensional space"  ''Teoret. Mat. Fiz.'' , '''42''' :  3  (1980)  pp. 343–349  (In Russian)</TD></TR></table>