TY - UNPB

T1 - On multi soliton solutions of the Sine-Gordon equation in more than one space dimension

AU - Zarmi, Yair

PY - 2013/4/1

Y1 - 2013/4/1

N2 - The (1+1)-dimensional Sine-Gordon equation passes integrability tests
commonly applied to nonlinear evolution equations. Its soliton solutions
are obtained by a Hirota algorithm. In higher space-dimensions, the
equation does not pass these tests. In this paper, using no more than
the relativistic kinematics of the tachyonic momentum vectors, from
which the soliton solutions are constructed through the Hirota
algorithm, the existence and classification of N-soliton solutions of
the (1+2)- and (1+3)-dimensional equations for all N greater than or
equal to 1 are presented. In (1+2) dimensions, each multisoliton
solution propagates rigidly at one velocity. The solutions are divided
into two subsets: Solutions whose velocities are lower than the speed of
light (c = 1), or are greater than or equal to c. In (1+3)-dimensions,
multisoliton solutions are characterized by spatial structure and
velocity composition. The spatial structure is either planar (rotated
(1+2)-dimensional solutions), or genuinely three-dimensional - branes.
Some solutions, planar or branes, propagate rigidly at one velocity,
which is lower than, equal to, or higher than c. A subset of the branes
contains hybrids, in which different clusters of solitons propagate at
different velocities. Some velocities may be lower than c but some must
be equal to, or greater than c. Finally, the speed of light cannot be
approached from within the subset of slower-than-light solutions in both
(1+2) and (1+3) dimensions.

AB - The (1+1)-dimensional Sine-Gordon equation passes integrability tests
commonly applied to nonlinear evolution equations. Its soliton solutions
are obtained by a Hirota algorithm. In higher space-dimensions, the
equation does not pass these tests. In this paper, using no more than
the relativistic kinematics of the tachyonic momentum vectors, from
which the soliton solutions are constructed through the Hirota
algorithm, the existence and classification of N-soliton solutions of
the (1+2)- and (1+3)-dimensional equations for all N greater than or
equal to 1 are presented. In (1+2) dimensions, each multisoliton
solution propagates rigidly at one velocity. The solutions are divided
into two subsets: Solutions whose velocities are lower than the speed of
light (c = 1), or are greater than or equal to c. In (1+3)-dimensions,
multisoliton solutions are characterized by spatial structure and
velocity composition. The spatial structure is either planar (rotated
(1+2)-dimensional solutions), or genuinely three-dimensional - branes.
Some solutions, planar or branes, propagate rigidly at one velocity,
which is lower than, equal to, or higher than c. A subset of the branes
contains hybrids, in which different clusters of solitons propagate at
different velocities. Some velocities may be lower than c but some must
be equal to, or greater than c. Finally, the speed of light cannot be
approached from within the subset of slower-than-light solutions in both
(1+2) and (1+3) dimensions.

KW - Nonlinear Sciences - Exactly Solvable and Integrable Systems

KW - 35Q51

KW - 37K40

KW - 35Q75

M3 - ???researchoutput.researchoutputtypes.workingpaper.preprint???

BT - On multi soliton solutions of the Sine-Gordon equation in more than one space dimension

ER -