901 research outputs found
A domain wall between single-mode and bimodal states and its transition to dynamical behavior in inhomogeneous systems
We consider domain walls (DW's) between single-mode and bimodal states that
occur in coupled nonlinear diffusion (NLD), real Ginzburg-Landau (RGL), and
complex Ginzburg-Landau (CGL) equations with a spatially dependent coupling
coefficient. Group-velocity terms are added to the NLD and RGL equations, which
breaks the variational structure of these models. In the simplest case of two
coupled NLD equations, we reduce the description of stationary configurations
to a single second-order ordinary differential equation. We demonstrate
analytically that a necessary condition for existence of a stationary DW is
that the group-velocity must be below a certain threshold value. Above this
threshold, dynamical behavior sets in, which we consider in detail. In the CGL
equations, the DW may generate spatio-temporal chaos, depending on the
nonlinear dispersion.Comment: 16 pages (latex) including 11 figures; accepted for publication in
Physica
Practical use of variational principles for modeling water waves
This paper describes a method for deriving approximate equations for
irrotational water waves. The method is based on a 'relaxed' variational
principle, i.e., on a Lagrangian involving as many variables as possible. This
formulation is particularly suitable for the construction of approximate water
wave models, since it allows more freedom while preserving the variational
structure. The advantages of this relaxed formulation are illustrated with
various examples in shallow and deep waters, as well as arbitrary depths. Using
subordinate constraints (e.g., irrotationality or free surface impermeability)
in various combinations, several model equations are derived, some being
well-known, other being new. The models obtained are studied analytically and
exact travelling wave solutions are constructed when possible.Comment: 30 pages, 1 figure, 62 references. Other author's papers can be
downloaded at http://www.denys-dutykh.com
Mobility of solitons in one-dimensional lattices with the cubic-quintic nonlinearity
We investigate mobility regimes for localized modes in the discrete nonlinear
Schr\"{o}dinger (DNLS) equation with the cubic-quintic onsite terms. Using the
variational approximation (VA), the largest soliton's total power admitting
progressive motion of kicked discrete solitons is predicted, by comparing the
effective kinetic energy with the respective Peierls-Nabarro (PN) potential
barrier. The prediction is novel for the DNLS model with the cubic-only
nonlinearity too, demonstrating a reasonable agreement with numerical findings.
Small self-focusing quintic term quickly suppresses the mobility. In the case
of the competition between the cubic self-focusing and quintic self-defocusing
terms, we identify parameter regions where odd and even fundamental modes
exchange their stability, involving intermediate asymmetric modes. In this
case, stable solitons can be set in motion by kicking, so as to let them pass
the PN barrier. Unstable solitons spontaneously start oscillatory or
progressive motion, if they are located, respectively, below or above a
mobility threshold. Collisions between moving discrete solitons, at the
competing nonlinearities frame, are studied too.Comment: 12 pages, 15 figure
Decomposition Method for Kdv Boussinesq and Coupled Kdv Boussinesq Equations
This paper obtains the solitary wave solutions of two different forms of Boussinesq equations that model the study of shallow water waves in lakes and ocean beaches. The decomposition method using He’s polynomials is applied to solve the governing equations. The travelling wave hypothesis is also utilized to solve the generalized case of coupled Boussinesq equations, and, thus, an exact soliton solution is obtained. The results are also supported by numerical simulations. Keywords: Decomposition Method, He’s polynomials, cubic Boussinesq equation, Coupled Boussinesq equation
Continuation for thin film hydrodynamics and related scalar problems
This chapter illustrates how to apply continuation techniques in the analysis
of a particular class of nonlinear kinetic equations that describe the time
evolution through transport equations for a single scalar field like a
densities or interface profiles of various types. We first systematically
introduce these equations as gradient dynamics combining mass-conserving and
nonmass-conserving fluxes followed by a discussion of nonvariational amendmends
and a brief introduction to their analysis by numerical continuation. The
approach is first applied to a number of common examples of variational
equations, namely, Allen-Cahn- and Cahn-Hilliard-type equations including
certain thin-film equations for partially wetting liquids on homogeneous and
heterogeneous substrates as well as Swift-Hohenberg and Phase-Field-Crystal
equations. Second we consider nonvariational examples as the
Kuramoto-Sivashinsky equation, convective Allen-Cahn and Cahn-Hilliard
equations and thin-film equations describing stationary sliding drops and a
transversal front instability in a dip-coating. Through the different examples
we illustrate how to employ the numerical tools provided by the packages
auto07p and pde2path to determine steady, stationary and time-periodic
solutions in one and two dimensions and the resulting bifurcation diagrams. The
incorporation of boundary conditions and integral side conditions is also
discussed as well as problem-specific implementation issues
Traveling Wave Solutions of ZK-BBM Equation Sine-Cosine Method
Travelling wave solutions are obtained by using a relatively new technique which is called sine-cosine method for ZK-BBM equations. Solution procedure and obtained results re-confirm the efficiency of the proposed scheme
Numerical simulation of the nonlinear coupled jaulent-miodek equation by elzaki transform-adomian polynomial method
The Elzaki transform which is an integral transform used to obtain solutions of linear differential equations is coupled with Adomian polynomial to solve nonlinear coupled Jaulent-Miodek (JM) equation. The Adomian polynomial is used to linearise the nonlinear functions in the partial differential equation before the scheme of the Elzaki transform was used to iteratively generate each term of the series solution. The solutions obtained were compared with the exact solutions and were found to give a very small error, the graphical representation of the solutions which give the shape of the solitons also agree with that of the Adomian decomposition method when a comparison is made. The method is powerful and effective as it does not involve large computer memory and does not involve discretizing the independent variables to achieve the required solution.publishedVersio
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