3,174 research outputs found
Conservation Laws, Symmetry Reductions, and New Exact Solutions of the (2 + 1)-Dimensional Kadomtsev-Petviashvili Equation with Time-Dependent Coefficients
The (2 + 1)-dimensional Kadomtsev-Petviashvili equation with time-dependent coefficients is investigated. By means of the Lie group method, we first obtain several geometric symmetries for the equation in terms of coefficient functions and arbitrary functions of t. Based on the obtained symmetries, many nontrivial and time-dependent conservation laws for the equation are obtained with the help of Ibragimov’s new conservation theorem. Applying the characteristic equations of the obtained symmetries, the (2 + 1)-dimensional KP equation is reduced to (1 + 1)-dimensional nonlinear partial differential equations, including a special case of (2 + 1)-dimensional Boussinesq equation and different types of the KdV equation. At the same time, many new exact solutions are derived such as soliton and soliton-like solutions and algebraically explicit analytical solutions
Exact accelerating solitons in nonholonomic deformation of the KdV equation with two-fold integrable hierarchy
Recently proposed nonholonomic deformation of the KdV equation is solved
through inverse scattering method by constructing AKNS-type Lax pair. Exact and
explicit N-soliton solutions are found for the basic field and the deforming
function showing an unusual accelerated (decelerated) motion. A two-fold
integrable hierarchy is revealed, one with usual higher order dispersion and
the other with novel higher nonholonomic deformations.Comment: 7 pages, 2 figures, latex. Exact explicit exact N-soliton solutions
(through ISM) for KdV field u and deforming function w are included. Version
to be published in J. Phys.
Group classification of variable coefficient KdV-like equations
The exhaustive group classification of the class of KdV-like equations with
time-dependent coefficients is carried out using
equivalence based approach. A simple way for the construction of exact
solutions of KdV-like equations using equivalence transformations is described.Comment: 8 pages; minor misprints are corrected. arXiv admin note: substantial
text overlap with arXiv:1104.198
Nonlinearizing linear equations to integrable systems including new hierarchies with nonholonomic deformations
We propose a scheme for nonlinearizing linear equations to generate
integrable nonlinear systems of both the AKNS and the KN classes, based on the
simple idea of dimensional analysis and detecting the building blocks of the
Lax pair. Along with the well known equations we discover a novel integrable
hierarchy of higher order nonholonomic deformations for the AKNS family, e.g.
for the KdV, the mKdV, the NLS and the SG equation, showing thus a two-fold
universality of the recently found deformation for the KdV equation.Comment: 17 pages, 5 figures, Latex, Final version to be published in J. Math.
Phy
Whitham modulation theory for the Kadomtsev-Petviashvili equation
The genus-1 KP-Whitham system is derived for both variants of the
Kadomtsev-Petviashvili (KP) equation (namely, the KPI and KPII equations). The
basic properties of the KP-Whitham system, including symmetries, exact
reductions, and its possible complete integrability, together with the
appropriate generalization of the one-dimensional Riemann problem for the
Korteweg-deVries equation are discussed. Finally, the KP-Whitham system is used
to study the linear stability properties of the genus-1 solutions of the KPI
and KPII equations; it is shown that all genus-1 solutions of KPI are linearly
unstable while all genus-1 solutions of KPII {are linearly stable within the
context of Whitham theory.Comment: Significantly revised versio
The Painleve Test and Reducibility to the Canonical Forms for Higher-Dimensional Soliton Equations with Variable-Coefficients
The general KdV equation (gKdV) derived by T. Chou is one of the famous (1+1)
dimensional soliton equations with variable coefficients. It is well-known that
the gKdV equation is integrable. In this paper a higher-dimensional gKdV
equation, which is integrable in the sense of the Painleve test, is presented.
A transformation that links this equation to the canonical form of the
Calogero-Bogoyavlenskii-Schiff equation is found. Furthermore, the form and
similar transformation for the higher-dimensional modified gKdV equation are
also obtained.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
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