Stability and Asymptotic Stability for Subcritical gKdV Equations

Martel, Yvan; Merle, Frank; Tsai, Tai‐Peng

doi:10.1007/s00220-002-0723-2

Cited by 210 publications

(329 citation statements)

References 22 publications

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“…In our main result, the logarithmic distance is due to strong attractive interaction between the two solitary waves. This is in contrast with most previous works on multi-solitary waves of (gKdV) where weak interactions do not change the behavior of solitons, see in particular [2], [4], [10], [15]. A typical example to illustrate weakly interacting dynamics is the existence of multi-soliton solutions u(t) of (gKdV) with any different speeds 0 < v 1 < ... < v K and any x 1 , ..., x K ∈ R,…”

Section: Resultscontrasting

confidence: 88%

“…We recall the coercivity property of L (see [15], [27]) in sub-critical cases: there exists µ > 0 such that for f ∈ H 1 (R),…”

Section: Resultsmentioning

confidence: 99%

“…(a) The proof of the coercivity property (3.20) is a standard consequence of (1.9) and the orthogonality properties (2.37) by an elementary localization argument. We refer to the proof of Lemma 4 in [15]. We observe that locally around each solitonR j , the functional behaves essentially asˆ(…”

Section: Proposition 7 (Coercivity and Time Control Of Energy Functiomentioning

confidence: 97%

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Strongly interacting multi-solitons with logarithmic relative distance for the gKdV equation

Vinh

2017

Nonlinearity

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Abstract. We consider the following class of equations of (gKdV) typewith mass sub-critical (2 < p < 5) and mass super-critical nonlinearities (p > 5). We prove the existence of 2-solitary wave solutions with logarithmic relative distance, i.e. solutions u(t) satisfyingwhere c = c(p) > 0 is a fixed constant, σ = −1 in sub-critical cases and σ = 1 in supercritical cases. For the integrable case (p = 3), such solution was known by integrability theory. This regime corresponds to strong attractive interactions. For sub-critical p, it was known that opposite sign traveling waves are attractive. For super-critical p, we derive from our computations that same sign traveling waves are attractive.

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Section: Resultscontrasting

confidence: 88%

“…We recall the coercivity property of L (see [15], [27]) in sub-critical cases: there exists µ > 0 such that for f ∈ H 1 (R),…”

Section: Resultsmentioning

confidence: 99%

Section: Proposition 7 (Coercivity and Time Control Of Energy Functiomentioning

confidence: 97%

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Strongly interacting multi-solitons with logarithmic relative distance for the gKdV equation

Vinh

2017

Nonlinearity

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“…In addition, from the information on u(T c ) and by asymptotic arguments ( [25], [21], [27] and [22]) related to sharp monotonicity properties, we fully describe the solution u(t) in large time, that is, for t > T .…”

Section: Resultsmentioning

confidence: 99%

Description of two soliton collision for the quartic gKdV equation

2011

Self Cite

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In this paper, we give the first description of the collision of two solitons for a nonintegrable equation in a special regime. We consider solutions of the quartic gKdV equation ∂tu + ∂x(∂ 2 x u + u 4 ) = 0, which behave aswhere Qc(x − ct) is a soliton and η(t) H 1 Qc 2 H 1 Qc 1 H 1 . The global behavior of u(t) is given by the following stability result: for all t ∈ R, u(t, x) = Q c 1 (t) (x − y1(t)) + Q c 2 (t) (x − y2(t)) + η(t, x), where η(t) H 1 Qc 2 H 1 and limt→+∞ c1(t) = c

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“…Here we are considering a different problem: stability of sums of 1-solitons (configurations which are not themselves solutions) in the energy space. Results of this type were obtained for KdV-type equations and NLS equations in [14,5,6] and [15], respectively. Our approach follows that of [14] for gKdV, which adds to the energy method of Weinstein [21] for the one soliton case, the monotonicity property of the L 2 -mass on the right of each soliton.…”

Section: 2)mentioning

confidence: 99%