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Using P. Lax’s concept of a Lax pair we prove global existence and uniqueness for solutions of the initial value problem for mKdV in classes of smooth functions which can be unbounded at infinity, and in particular, may tend to i...
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Using P. Lax’s concept of a Lax pair we prove global existence and uniqueness for solutions of the initial value problem for mKdV in classes of smooth functions which can be unbounded at infinity, and in particular, may tend to infinity with respect to the space variable. Moreover, we establish the invariance of the spectrum and the unitary type of the Schrodinger operator under the KdV flow and the invariance of the spectrum and the unitary type of the impedance operator under the mKdV flow for potentials in these classes.
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We point out that the N = 4 supersymmetric KdV hierarchy, when written through the prepotentials of the bosonic chiral and antichiral N = 2 supercurrents, exhibits a freedom related to the possibility to choose different gauges fo...
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We point out that the N = 4 supersymmetric KdV hierarchy, when written through the prepotentials of the bosonic chiral and antichiral N = 2 supercurrents, exhibits a freedom related to the possibility to choose different gauges for the prepotentials. Some implications of this property are presented. In particular, we give the prepotential form of the "small" N = 4 superconformal algebra, the second Hamiltonian structure algebra of the N = 4 SKdV hierarchy, for two choices of gauge. [References: 9]
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In this paper we discuss properties of the Korteweg–de Vries (KdV) equation under periodic boundary conditions, especially those which are important for studying perturbations of the equation. We then review what is known about t...
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In this paper we discuss properties of the Korteweg–de Vries (KdV) equation under periodic boundary conditions, especially those which are important for studying perturbations of the equation. We then review what is known about the long-time behaviour of solutions for perturbed KdV equations.
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The second-order Korteweg–de Vries (sKdV) equation was introduced as a type of KdV-typed model that describes the wave propagations in a weakly nonlinear and weakly dispersive system. However, the question about the multisoliton ...
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The second-order Korteweg–de Vries (sKdV) equation was introduced as a type of KdV-typed model that describes the wave propagations in a weakly nonlinear and weakly dispersive system. However, the question about the multisoliton solution still remains open. In this article, we discovered numerically that the solitons with different speeds and amplitudes seem to be almost unaffected in shapes by passing through each other (though this could cause a change in their position). Such a double-soliton phenomenon characterizes the most important feature of the equation. In addition, we present the conservation laws, Hamiltonian and Lagrangian density functions for the equation and perform the numerical computation to shed light on the existence of soliton phenomenon.
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For a dusty plasma with a negatively charged dust fluid with Boltzmann distributed electrons and superthermal ions, the dust acoustic solitary waves have been studied in this paper. We derived a Korteweg-de Vries (KdV) equation an...
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For a dusty plasma with a negatively charged dust fluid with Boltzmann distributed electrons and superthermal ions, the dust acoustic solitary waves have been studied in this paper. We derived a Korteweg-de Vries (KdV) equation and a modified KdV equation for different cases. It was shown that the superthermal distributed ion have very important effect on the characters of dust acoustic solitary waves.
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A class of linear partial differential equations whose coefficients are solutions of nonlinear integrable partial differential equations can be solved by a generalized inverse scattering method. The method applies either in the ca...
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A class of linear partial differential equations whose coefficients are solutions of nonlinear integrable partial differential equations can be solved by a generalized inverse scattering method. The method applies either in the case of initial conditions on an x or t axis or in the case of boundary conditions in the quarterplane x >= 0, t >= 0. Some of the solved equations are of physical interest.
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We consider a Boussinesq system of KdV-KdV type introduced by J.L. Bona, M. Chen and J.-C. Saut as a model for the motion of small amplitude long waves on the surface of an ideal fluid. This system of two equations can describe th...
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We consider a Boussinesq system of KdV-KdV type introduced by J.L. Bona, M. Chen and J.-C. Saut as a model for the motion of small amplitude long waves on the surface of an ideal fluid. This system of two equations can describe the propagation of waves in both directions, while the single KdV equation is limited to unidirectional waves. We are concerned here with the exact controllability of the Boussinesq system by using some boundary controls. By reducing the controllability problem to a spectral problem which is solved by using the Paley-Wiener method introduced by the third author for KdV, we determine explicitly all the critical lengths for which the exact controllability fails for the linearized system, and give a complete picture of the controllability results with one or two boundary controls of Dirichlet or Neumann type. The extension of the exact controllability to the full Boussinesq system is derived in the energy space in the case of a control of Neumann type. It is obtained by incorporating a boundary feedback in the control in order to ensure a global Kato smoothing effect.
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A triple-humped soliton equation is considered. A corresponding Backlund transformation for it is obtained and a nonlinear superposition formula is presented. As an application of the results obtained, soliton solutions first foun...
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A triple-humped soliton equation is considered. A corresponding Backlund transformation for it is obtained and a nonlinear superposition formula is presented. As an application of the results obtained, soliton solutions first found by Narita to the system are: rederived. A sequence of rational solutions are found and other types of solutions which are a mix of exponentials and rational expressions are deduced. [References: 8]
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The coupled Kadomtsev-Petviashvili (KP) equations with variable coefficients and Wick-type stochastic coupled KP equations are investigated. Exact solutions are shown using the Hermite transform, the homogeneous balance principle ...
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The coupled Kadomtsev-Petviashvili (KP) equations with variable coefficients and Wick-type stochastic coupled KP equations are investigated. Exact solutions are shown using the Hermite transform, the homogeneous balance principle and the F-expansion method.
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Employing Hirota's method, a class of soliton solutions for the N = 2 super mKdV equations is proposed in terms of a single Grassmann parameter. Such solutions are shown to satisfy two copies of N = 1 supersymmetric mKdV equations...
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Employing Hirota's method, a class of soliton solutions for the N = 2 super mKdV equations is proposed in terms of a single Grassmann parameter. Such solutions are shown to satisfy two copies of N = 1 supersymmetric mKdV equations connected by nontrivial algebraic identities. Using the super Miura transformation, we obtain solutions of the N = 2 super KdV equations. These are shown to generalize solutions derived previously. By using them KdV/sinh-Gordon hierarchy properties we generate the solutions of the N = 2 super sinh-Gordon as well.
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