Integrable and solvable systems, and relations among them

There are several different classes of differential equations that may be described as ‘integrable’ or ‘solvable’. For example, there are completely integrable dynamical systems; equations such as the sine—Gordon equation, which admit soliton solutions; and the self-dual gauge-field equations in four dimensions (with generalizations in arbitrarily large dimension). This lecture discusses two ideas that link all of these together: one is the Painlevé property, which says (roughly speaking) that all solutions to the equations are meromorphic; the other is that many of the equations are special cases (i.e. reductions) of others.

Proc. R. Soc. Lond . A 359, 479-495 (1978) Exact, multiple soliton solutions of the double sine Gordon equation By P. G. Burt Equation (A 10), exponent of P should read - 1/(2 p ). Equation (A 15), exponents of P should read - 1/(2 p ) and - (1/(2 p ) + 1) respectively. Proc. R. Soc. Lond . A 362, 281-303 (1978) The Bakerian Lecture, 1977: in vitro models for photosynthesis By Sir George Porter, F. R. S. Page 300, line —8: for λ < 455 nm read λ > 455 nm


2020 ◽  
Vol 2020 (6) ◽  
Author(s):  
Y Hanif ◽  
U Saleem

Abstract We study the discrete Darboux transformation and construct multi-soliton solutions in terms of the ratio of determinants for the integrable discrete sine-Gordon equation. We also calculate explicit expressions of single-, double-, triple-, and quadruple-soliton solutions as well as single- and double-breather solutions of the discrete sine-Gordon equation. The dynamical features of discrete kinks and breathers are also illustrated.


1994 ◽  
Vol 06 (06) ◽  
pp. 1301-1338 ◽  
Author(s):  
W. OEVEL ◽  
W. SCHIEF

It is shown that products of eigenfunctions and (integrated) adjoint eigenfunctions associated with the (modified) Kadomtsev-Petviashvili (KP) hierarchy form generators of a symmetry transformation. Linear integro-differential representations for these symmetries are found. For special cases the corresponding nonlinear equations are the compatibility conditions of linear scattering problems of Loewner type. The examples include the 2+1-dimensional sine-Gordon equation with space variables occuring on an equal footing introduced recently by Konopelchenko and Rogers. This equation represents a special squared eigenfunction symmetry of the Ishimori hierarchy.


2009 ◽  
Vol 23 (04) ◽  
pp. 607-621 ◽  
Author(s):  
SONG-LIN ZHAO ◽  
HONG-HAI HAO ◽  
DA-JUN ZHANG

We derive multi-soliton solutions for a non-isospectral mKdV–sine-Gordon equation by means of a bilinear approach. Solutions are given in both Hirota's form and Wronskian form. This non-isospectral equation is a generic one which is related to a time-dependent spectral parameter with time evolution [Formula: see text].


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