The Experts below are selected from a list of 11238 Experts worldwide ranked by ideXlab platform
Gerald Teschl - One of the best experts on this subject based on the ideXlab platform.
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rarefaction waves of the korteweg de Vries Equation via nonlinear steepest descent
Journal of Differential Equations, 2016Co-Authors: Kyrylo Andreiev, Iryna Egorova, Till Luc Lange, Gerald TeschlAbstract:Abstract We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the Korteweg–de Vries Equation with steplike initial data leading to a rarefaction wave. In addition to the leading asymptotic we also compute the next term in the asymptotic expansion of the rarefaction wave, which was not known before.
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on the cauchy problem for the korteweg de Vries Equation with steplike finite gap initial data ii perturbations with finite moments
Journal D Analyse Mathematique, 2011Co-Authors: Iryna Egorova, Gerald TeschlAbstract:We solve the Cauchy problem for the Korteweg-De Vries Equation with steplike quasi-periodic, finite-gap initial conditions under the assumption that the perturbations have a given number of finite derivatives with finite moments.
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on the cauchy problem for the korteweg de Vries Equation with steplike finite gap initial data ii perturbations with finite moments
arXiv: Exactly Solvable and Integrable Systems, 2009Co-Authors: Iryna Egorova, Gerald TeschlAbstract:We solve the Cauchy problem for the Korteweg-De Vries Equation with steplike quasi-periodic, finite-gap initial conditions under the assumption that the perturbations have a given number of derivatives with finite moments.
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long time asymptotics for the korteweg de Vries Equation via nonlinear steepest descent
Mathematical Physics Analysis and Geometry, 2009Co-Authors: Katrin Grunert, Gerald TeschlAbstract:We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the Korteweg-De Vries Equation for decaying initial data in the soliton and similarity region. This paper can be viewed as an expository introduction to this method.
Ying Liu - One of the best experts on this subject based on the ideXlab platform.
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soliton management for a variable coefficient modified korteweg de Vries Equation
Physical Review E, 2011Co-Authors: Zhiyuan Sun, Yi-tian Gao, Ying LiuAbstract:The concept of soliton management has been explored in the Bose-Einstein condensate and optical fibers. In this paper, our purpose is to investigate whether a similar concept exists for a variable-coefficient modified Korteweg--de Vries Equation, which arises in the interfacial waves in two-layer liquid and Alfv\'en waves in a collisionless plasma. Through the Painlev\'e test, a generalized integrable form of such an Equation has been constructed under the Painlev\'e constraints of the variable coefficients based on the symbolic computation. By virtue of the Ablowitz-Kaup-Newell-Segur system, a Lax pair with time-dependent nonisospectral flow of the integrable form has been established under the Lax constraints which appear to be more rigid than the Painlev\'e ones. Under such Lax constraints, multisoliton solutions for the completely integrable variable-coefficient modified Korteweg--de Vries Equation have been derived via the Hirota bilinear method. Moreover, results show that the solitons and breathers with desired amplitude and width can be derived via the different choices of the variable coefficients.
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solitonic propagation and interaction for a generalized variable coefficient forced korteweg de Vries Equation in fluids
Physical Review E, 2011Co-Authors: Yi-tian Gao, Zhiyuan Sun, Ying LiuAbstract:Under investigation is a generalized variable-coefficient forced Korteweg-De Vries Equation in fluids and other fields. From the bilinear form of such Equation, the N-soliton solution and a type of analytic solution are constructed with symbolic computation. Analytic analysis indicates that: (1) dispersive and dissipative coefficients affect the solitonic velocity; (2) external-force term affects the solitonic velocity and background; (3) line-damping coefficient and some parameters affect the solitonic velocity, background, and amplitude. Solitonic propagation and interaction can be regarded as the combination of the effects of various variable coefficients. According to a constraint among the nonlinear, dispersive, and line-damping coefficients in this paper, the possible applications of our results in the real world are also discussed in three aspects, i.e., solution with the constraint, solution without the constraint, and approximate solution.
Pierre Raphael - One of the best experts on this subject based on the ideXlab platform.
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blow up for the critical generalized korteweg de Vries Equation i dynamics near the soliton
Acta Mathematica, 2014Co-Authors: Yvan Martel, Frank Merle, Pierre RaphaelAbstract:We consider the quintic generalized Korteweg–de Vries Equation (gKdV) $$u_t + (u_{xx} + u^5)_x =0,$$ which is a canonical mass critical problem, for initial data in H1 close to the soliton. In earlier works on this problem, finite- or infinite-time blow up was proved for non-positive energy solutions, and the solitary wave was shown to be the universal blow-up profile, see [16], [26] and [20]. For well-localized initial data, finite-time blow up with an upper bound on blow-up rate was obtained in [18].
Yi-tian Gao - One of the best experts on this subject based on the ideXlab platform.
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soliton management for a variable coefficient modified korteweg de Vries Equation
Physical Review E, 2011Co-Authors: Zhiyuan Sun, Yi-tian Gao, Ying LiuAbstract:The concept of soliton management has been explored in the Bose-Einstein condensate and optical fibers. In this paper, our purpose is to investigate whether a similar concept exists for a variable-coefficient modified Korteweg--de Vries Equation, which arises in the interfacial waves in two-layer liquid and Alfv\'en waves in a collisionless plasma. Through the Painlev\'e test, a generalized integrable form of such an Equation has been constructed under the Painlev\'e constraints of the variable coefficients based on the symbolic computation. By virtue of the Ablowitz-Kaup-Newell-Segur system, a Lax pair with time-dependent nonisospectral flow of the integrable form has been established under the Lax constraints which appear to be more rigid than the Painlev\'e ones. Under such Lax constraints, multisoliton solutions for the completely integrable variable-coefficient modified Korteweg--de Vries Equation have been derived via the Hirota bilinear method. Moreover, results show that the solitons and breathers with desired amplitude and width can be derived via the different choices of the variable coefficients.
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solitonic propagation and interaction for a generalized variable coefficient forced korteweg de Vries Equation in fluids
Physical Review E, 2011Co-Authors: Yi-tian Gao, Zhiyuan Sun, Ying LiuAbstract:Under investigation is a generalized variable-coefficient forced Korteweg-De Vries Equation in fluids and other fields. From the bilinear form of such Equation, the N-soliton solution and a type of analytic solution are constructed with symbolic computation. Analytic analysis indicates that: (1) dispersive and dissipative coefficients affect the solitonic velocity; (2) external-force term affects the solitonic velocity and background; (3) line-damping coefficient and some parameters affect the solitonic velocity, background, and amplitude. Solitonic propagation and interaction can be regarded as the combination of the effects of various variable coefficients. According to a constraint among the nonlinear, dispersive, and line-damping coefficients in this paper, the possible applications of our results in the real world are also discussed in three aspects, i.e., solution with the constraint, solution without the constraint, and approximate solution.
Yvan Martel - One of the best experts on this subject based on the ideXlab platform.
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blow up for the critical generalized korteweg de Vries Equation i dynamics near the soliton
Acta Mathematica, 2014Co-Authors: Yvan Martel, Frank Merle, Pierre RaphaelAbstract:We consider the quintic generalized Korteweg–de Vries Equation (gKdV) $$u_t + (u_{xx} + u^5)_x =0,$$ which is a canonical mass critical problem, for initial data in H1 close to the soliton. In earlier works on this problem, finite- or infinite-time blow up was proved for non-positive energy solutions, and the solitary wave was shown to be the universal blow-up profile, see [16], [26] and [20]. For well-localized initial data, finite-time blow up with an upper bound on blow-up rate was obtained in [18].
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a liouville theorem for the critical generalized korteweg de Vries Equation
Journal de Mathématiques Pures et Appliquées, 2000Co-Authors: Yvan Martel, Frank MerleAbstract:Abstract We prove in this paper a rigidity theorem on the flow of the critical generalized Korteweg–de Vries Equation close to a soliton up to scaling and translation. To prove this result we introduce new tools to understand nonlinear phenomenon. This will give a result of asymptotic completeness.
