22 research outputs found
Lie symmetries, conservation laws and exact solutions of a generalized quasilinear KdV equation with degenerate dispersion
Mathematics Subject Classi cation. Primary: 35B06, 35L65, 35C07; Secondary: 35Q53.We provide a complete classification of point symmetries and low-order local conservation laws of the generalized quasilinear KdV equation in terms of the arbitrary function. The corresponding interpretation of symmetry transformation groups are given. In addition, a physical description of the conserved quantities is included. Finally, few travelling wave solutions have been obtained.11 página
Superposition of Elliptic Functions as Solutions For a Large Number of Nonlinear Equations
For a large number of nonlinear equations, both discrete and continuum, we
demonstrate a kind of linear superposition. We show that whenever a nonlinear
equation admits solutions in terms of both Jacobi elliptic functions \cn(x,m)
and \dn(x,m) with modulus , then it also admits solutions in terms of
their sum as well as difference. We have checked this in the case of several
nonlinear equations such as the nonlinear Schr\"odinger equation, MKdV, a mixed
KdV-MKdV system, a mixed quadratic-cubic nonlinear Schr\"odinger equation, the
Ablowitz-Ladik equation, the saturable nonlinear Schr\"odinger equation,
, the discrete MKdV as well as for several coupled field
equations. Further, for a large number of nonlinear equations, we show that
whenever a nonlinear equation admits a periodic solution in terms of
\dn^2(x,m), it also admits solutions in terms of \dn^2(x,m) \pm \sqrt{m}
\cn(x,m) \dn(x,m), even though \cn(x,m) \dn(x,m) is not a solution of these
nonlinear equations. Finally, we also obtain superposed solutions of various
forms for several coupled nonlinear equations.Comment: 40 pages, no figure
Spectral theory of soliton and breather gases for the focusing nonlinear Schrödinger equation
Solitons and breathers are localized solutions of integrable systems that can be viewed as “particles” of complex statistical objects called soliton and breather gases. In view of the growing evidence of their ubiquity in fluids and nonlinear optical media, these “integrable” gases present a fundamental interest for nonlinear physics. We develop an analytical theory of breather and soliton gases by considering a special, thermodynamic-type limit of the wave-number–frequency relations for multiphase (finite-gap) solutions of the focusing nonlinear Schrödinger equation. This limit is defined by the locus and the critical scaling of the band spectrum of the associated Zakharov-Shabat operator, and it yields the nonlinear dispersion relations for a spatially homogeneous breather or soliton gas, depending on the presence or absence of the “background” Stokes mode. The key quantity of interest is the density of states defining, in principle, all spectral and statistical properties of a soliton (breather) gas. The balance of terms in the nonlinear dispersion relations determines the nature of the gas: from an ideal gas of well separated, noninteracting breathers (solitons) to a special limiting state, which we term a breather (soliton) condensate, and whose properties are entirely determined by the pairwise interactions between breathers (solitons). For a nonhomogeneous breather gas, we derive a full set of kinetic equations describing the slow evolution of the density of states and of its carrier wave counterpart. The kinetic equation for soliton gas is recovered by collapsing the Stokes spectral band. A number of concrete examples of breather and soliton gases are considered, demonstrating the efficacy of the developed general theory with broad implications for nonlinear optics, superfluids, and oceanography. In particular, our work provides the theoretical underpinning for the recently observed remarkable connection of the soliton gas dynamics with the long-term evolution of spontaneous modulational instability
Stability of smooth solitary waves for the generalized Korteweg - de Vries equation with combined dispersion
The orbital stability problem of the smooth solitary waves in the generalized Korteweg - de Vries equation with combined dispersion is considered. The results show that the smooth solitary waves are stable for an arbitrary speed of wave propagation.Розглянуто задачу про орбітальну стійкість гладких відокремлених хвиль для узагальненого рівняння Кортевега – де Фріза з комбінованою дисперсією. Отримані результати показують, що гладкі відокремлені хвилі є стійкими при довільній швидкості поширення хвиль
The higher grading structure of the WKI hierarchy and the two-component short pulse equation
A higher grading affine algebraic construction of integrable hierarchies,
containing the Wadati-Konno-Ichikawa (WKI) hierarchy as a particular case, is
proposed. We show that a two-component generalization of the Sch\" afer-Wayne
short pulse equation arises quite naturally from the first negative flow of the
WKI hierarchy. Some novel integrable nonautonomous models are also proposed.
The conserved charges, both local and nonlocal, are obtained from the Riccati
form of the spectral problem. The loop-soliton solutions of the WKI hierarchy
are systematically constructed through gauge followed by reciprocal B\" acklund
transformation, establishing the precise connection between the whole WKI and
AKNS hierarchies. The connection between the short pulse equation with the
sine-Gordon model is extended to a correspondence between the two-component
short pulse equation and the Lund-Regge model
Discretizations of the generalized AKNS scheme
We consider space discretizations of the matrix Zakharov-Shabat AKNS scheme,
in particular the discrete matrix non-linear Scrhr\"odinger (DNLS) model, and
the matrix generalization of the Ablowitz-Ladik (AL) model, which is the more
widely acknowledged discretization. We focus on the derivation of solutions via
local Darboux transforms for both discretizations, and we derive novel
solutions via generic solutions of the associated discrete linear equations.
The continuum analogue is also discussed, and as an example we identify
solutions of the matrix NLS equation in terms of the heat kernel. In this frame
we also derive a discretization of the Burgers equation via the analogue of the
Cole-Hopf transform. Using the basic Darboux transforms for each scheme we
identify both matrix DNLS-like and AL hierarchies, i.e. we extract the
associated Lax pairs, via the dressing process. We also discuss the global
Darboux transform, which is the discrete analogue of the integral transform,
through the discrete Gelfand-Levitan-Marchenko (GLM) equation. The derivation
of the discrete matrix GLM equation and associated solutions are also presented
together with explicit linearizations. Particular emphasis is given in the
discretization schemes, i.e. forward/backward in the discrete matrix DNLS
scheme versus symmetric in the discrete matrix AL model.Comment: 28 pages, Latex. Typos corrected and clarifying comments added.
Version accepted in J. Phys.