713 research outputs found
Construction of multi-solitons for the energy-critical wave equation in dimension 5
We construct 2-solitons of any speed of the focusing energy-critical
nonlinear wave equation in dimension 5. The existence result also holds for the
case of K-solitons, for any K >2, assuming that the speeds are collinear. The
main difficulty of the construction is the strong interaction between the
solitons due to the slow spatial decay of the single soliton. This is in
contrast with previous constructions of multi-solitons for other nonlinear
models (like generalized KdV and nonlinear Schrodinger equations in energy
subcritical cases), where the interactions are exponentially small in time due
to the exponential decay of the solitons
Inelastic interaction of nearly equal solitons for the BBM equation
This paper is concerned with the interaction of two solitons of nearly equal
speeds for the (BBM) equation. This work is an extension of the results
obtained in arXiv:0910.3204 by the same authors, addressing the same question
for the quartic (gKdV) equation. First, we prove that the two solitons are
preserved by the interaction and that for all time they are separated by a
large distance, as in the case of the integrable (KdV) equation in this regime.
Second, we prove that the collision is not perfectly elastic, except in the
integrable case (i.e. in the limiting case of the (KdV) equation)
Description of two soliton collision for the quartic gKdV equation
This paper concerns the problem of collision of two solitons for the quartic
generalized Korteweg-de Vries equation. We introduce a new framework to
describe the collision in the special case where one soliton is small with
respect to the other. We prove that the two soliton survive the collision, we
describe the collision phenomenon (computation of the first order of the
resulting shifts on the solitons). Moreover, we prove that in this situation,
there does not exist pure two-soliton solutions
On the nonexistence of pure multi-solitons for the quartic gKdV equation
We consider the quartic (nonintegrable) (gKdV) equation. Let u(t) be an
outgoing 2-soliton of the equation, i.e. a solution behaving exactly as the sum
of two solitons (of speeds c1 and c2) for large positive time.
In arXiv:0910.3204, for nearly equal solitons, the solution u(t) is computed
up to some order of epsilon=1-c2/c1, everywhere in time and space. In
particular, it is deduced that u(t) is not a multi-soliton for large negative
time, proving the nonexistence of pure multi-soliton in this context.
In the present paper, we prove the same result for an explicit range of
speeds: 3/4 c1< c2< c1, by a different approach, which does not longer require
a precise description of the solution. In fact, the nonexistence result holds
for outgoing N-solitons, for any N>1, under an explicit assumption on the
speeds, which is a natural generalization of the condition for N=2.Comment: to appear in Int Math Res Notice
Refined asymptotics around solitons for gKdV equations
We consider the generalized Korteweg-de Vries equation
with general nonlinearity . Under an explicit condition on and
, there exists a solution in the energy space of the type
, called soliton. Stability theory for is
well-known.
In previous works, we have proved that for , , the family
of solitons is asymptotically stable in some local sense in , i.e. if
is close to (for all ), then locally
converges in the energy space to some as , for some
.
Then, the asymptotic stability result could be extended to the case of
general assumptions on and .
The objective of this paper is twofold.
The main objective is to prove that in the case , ,
has limit as under the additional assumption
.
The second objective of this paper is to provide large time stability and
asymptotic stability results for two soliton solutions for the case of general
nonlinearity , when the ratio of the speeds of the solitons is small. The
motivation is to accompany forthcoming works devoted to the collision of two
solitons in the nonintegrable case. The arguments are refinements of previous
works specialized to the case , for .Comment: Minor changes. To appear in Discrete and Continuous Dynamical Systems
- Series
- …