TY - UNPB

T1 - Sine-Gordon equation in higher dimensions: A fresh look at integrability

AU - Zarmi, Yair

PY - 2014/5/1

Y1 - 2014/5/1

N2 - The Sine-Gordon equation is integrable in (1+1)-dimensional Minkowski
and in 2-dimensional Euclidean spaces. In each case, it has a Lax pair,
and a Hirota algorithm generates its N soliton solutions for all N
greater than or equal to 1. The (1+2)-dimensional equation does not pass
known integrability tests and does not have a Lax pair. Still, the
Hirota algorithm generates N soliton solutions of that equation for all
N greater than or equal to 1. Each multi-soliton solution propagates
rigidly at a constant velocity, v. The solutions are divided into two
unconnected subspaces: Solutions with v greater than or equal to c =1,
and v smaller than c. Each subspace is connected by an invertible
transformation (rotation plus dilation) to the space of soliton
solutions of an integrable Sine-Gordon equation in two dimensions. The
faster-than-light solutions are connected to the solutions in
(1+1)-dimensional Minkowski space. The slower-than-light solutions are
connected to the solutions in 2-dimensional Euclidean space by this
transformation and also by Lorentz transformations. The Sine-Gordon
equation in (1+3)-dimensional Minkowski space has a richer variety of
solutions. Its slower-than-light solutions are connected to the
solutions of the integrable equation in 2-dimensional Euclidean space.
However, only a subset of its faster-than-light solutions is connected
to the solutions of the integrable equation in (1+1)-dimensional
Minkowski space.

AB - The Sine-Gordon equation is integrable in (1+1)-dimensional Minkowski
and in 2-dimensional Euclidean spaces. In each case, it has a Lax pair,
and a Hirota algorithm generates its N soliton solutions for all N
greater than or equal to 1. The (1+2)-dimensional equation does not pass
known integrability tests and does not have a Lax pair. Still, the
Hirota algorithm generates N soliton solutions of that equation for all
N greater than or equal to 1. Each multi-soliton solution propagates
rigidly at a constant velocity, v. The solutions are divided into two
unconnected subspaces: Solutions with v greater than or equal to c =1,
and v smaller than c. Each subspace is connected by an invertible
transformation (rotation plus dilation) to the space of soliton
solutions of an integrable Sine-Gordon equation in two dimensions. The
faster-than-light solutions are connected to the solutions in
(1+1)-dimensional Minkowski space. The slower-than-light solutions are
connected to the solutions in 2-dimensional Euclidean space by this
transformation and also by Lorentz transformations. The Sine-Gordon
equation in (1+3)-dimensional Minkowski space has a richer variety of
solutions. Its slower-than-light solutions are connected to the
solutions of the integrable equation in 2-dimensional Euclidean space.
However, only a subset of its faster-than-light solutions is connected
to the solutions of the integrable equation in (1+1)-dimensional
Minkowski space.

KW - Nonlinear Sciences - Exactly Solvable and Integrable Systems

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BT - Sine-Gordon equation in higher dimensions: A fresh look at integrability

ER -