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GU Chaohao The soliton theory is an important branch of nonlinear
science. On one hand, it describes various kinds of stable motions
appearing in - ture, such as solitary water wave, solitary signals
in optical ?bre etc., and has many applications in science and
technology (like optical signal communication). On the other hand,
it gives many e?ective methods ofgetting explicit solutions of
nonlinear partial di?erential equations. Therefore, it has
attracted much attention from physicists as well as mathematicians.
Nonlinearpartialdi?erentialequationsappearinmanyscienti?cpr- lems.
Getting explicit solutions is usually a di?cult task. Only in c-
tain special cases can the solutions be written down explicitly.
However, for many soliton equations, people have found quite a few
methods to get explicit solutions. The most famous ones are the
inverse scattering method, B] acklund transformation etc.. The
inverse scattering method is based on the spectral theory of
ordinary di?erential equations. The
Cauchyproblemofmanysolitonequationscanbetransformedtosolving a
system of linear integral equations. Explicit solutions can be
derived when the kernel of the integral equation is degenerate. The
B] ac ] klund transformation gives a new solution from a known
solution by solving a system of completely integrable partial
di?erential equations. Some complicated "nonlinear superposition
formula" arise to substitute the superposition principlein linear
science."
GU Chaohao The soliton theory is an important branch of nonlinear
science. On one hand, it describes various kinds of stable motions
appearing in - ture, such as solitary water wave, solitary signals
in optical ?bre etc., and has many applications in science and
technology (like optical signal communication). On the other hand,
it gives many e?ective methods ofgetting explicit solutions of
nonlinear partial di?erential equations. Therefore, it has
attracted much attention from physicists as well as mathematicians.
Nonlinearpartialdi?erentialequationsappearinmanyscienti?cpr- lems.
Getting explicit solutions is usually a di?cult task. Only in c-
tain special cases can the solutions be written down explicitly.
However, for many soliton equations, people have found quite a few
methods to get explicit solutions. The most famous ones are the
inverse scattering method, B] acklund transformation etc.. The
inverse scattering method is based on the spectral theory of
ordinary di?erential equations. The
Cauchyproblemofmanysolitonequationscanbetransformedtosolving a
system of linear integral equations. Explicit solutions can be
derived when the kernel of the integral equation is degenerate. The
B] ac ] klund transformation gives a new solution from a known
solution by solving a system of completely integrable partial
di?erential equations. Some complicated "nonlinear superposition
formula" arise to substitute the superposition principlein linear
science."
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