scholarly journals GBDT version of the Darboux transformation for the matrix coupled dispersionless equations (local and non-local cases)

2020 ◽  
Vol 5 (1) ◽  
Author(s):  
Roman O Popovych ◽  
Alexander L Sakhnovich
Keyword(s):  
Darboux Transformation ◽  
The Matrix ◽  
Non Local ◽  
Explicit Expressions

Abstract We introduce matrix coupled (local and non-local) dispersionless equations, construct GBDT (generalized Bäcklund-Darboux transformation) for these equations, derive wide classes of explicit multipole solutions, give explicit expressions for the corresponding Darboux and wave matrix valued functions and study their asymptotics in some interesting cases. We consider the scalar cases of coupled, complex coupled and non-local dispersionless equations as well.

scholarly journals Darboux transformation and multi-soliton solutions of the discrete sine-Gordon equation

2020 ◽  
Vol 2020 (6) ◽  
Author(s):  
Y Hanif ◽  
U Saleem
Keyword(s):  
Darboux Transformation ◽  
Gordon Equation ◽  
Soliton Solutions ◽  
Sine Gordon Equation ◽  
Dynamical Features ◽  
Discrete Darboux Transformation ◽  
Explicit Expressions

Abstract We study the discrete Darboux transformation and construct multi-soliton solutions in terms of the ratio of determinants for the integrable discrete sine-Gordon equation. We also calculate explicit expressions of single-, double-, triple-, and quadruple-soliton solutions as well as single- and double-breather solutions of the discrete sine-Gordon equation. The dynamical features of discrete kinks and breathers are also illustrated.

scholarly journals Matrix basis for plane and modal waves in a Timoshenko beam

2016 ◽  
Vol 3 (11) ◽  
pp. 160825 ◽  
Author(s):  
Julio Cesar Ruiz Claeyssen ◽  
Daniela de Rosso Tolfo ◽  
Leticia Tonetto
Keyword(s):  
Closed Form ◽  
Timoshenko Beam ◽  
Plane Waves ◽  
Crack Problem ◽  
Wave Interaction ◽  
Beam Model ◽  
High Frequencies ◽  
Transmitted Waves ◽  
The Matrix ◽  
Non Local

Plane waves and modal waves of the Timoshenko beam model are characterized in closed form by introducing robust matrix basis that behave according to the nature of frequency and wave or modal numbers. These new characterizations are given in terms of a finite number of coupling matrices and closed form generating scalar functions. Through Liouville’s technique, these latter are well behaved at critical or static situations. Eigenanalysis is formulated for exponential and modal waves. Modal waves are superposition of four plane waves, but there are plane waves that cannot be modal waves. Reflected and transmitted waves at an interface point are formulated in matrix terms, regardless of having a conservative or a dissipative situation. The matrix representation of modal waves is used in a crack problem for determining the reflected and transmitted matrices. Their euclidean norms are seen to be dominated by certain components at low and high frequencies. The matrix basis technique is also used with a non-local Timoshenko model and with the wave interaction with a boundary. The matrix basis allows to characterize reflected and transmitted waves in spectral and non-spectral form.

The order-n breather and degenerate breather solutions of the (2+1)-dimensional cmKdV equations

2021 ◽  
Vol 35 (04) ◽  
pp. 2150053
Author(s):  
Feng Yuan
Keyword(s):  
Plane Wave ◽  
Darboux Transformation ◽  
Taylor Expansion ◽  
Degradation Process ◽  
Dynamic Evolution ◽  
Breather Solution ◽  
De Vries ◽  
Evolution With Time ◽  
Explicit Expressions

Starting with a plane wave seed, the order-[Formula: see text] breather for the (2+1)-D complex modified Korteweg-de Vries (cmKdV) equations is obtained by the use of Darboux transformation. The dynamic evolution of order-2 and order-3 breather solutions is shown in the form of pictures. Afterward, we obtain the order-[Formula: see text] degenerate breather solution by using the Taylor expansion concerning the limits [Formula: see text] and focus on the order-2 degenerate breather solution. We show the dynamic evolution with time and discuss the degradation process from a breather solution through getting [Formula: see text] closer and closer to [Formula: see text]. Furthermore, the approximate trajectories of the order-2, order-3, order-4 degenerate breather solutions are depicted by explicit expressions, respectively.

Darboux transformation and multi-soliton solutions of local/nonlocal N-wave interactions

2017 ◽  
Vol 32 (36) ◽  
pp. 1750196 ◽  
Author(s):  
H. Sarfraz ◽  
U. Saleem
Keyword(s):  
Darboux Transformation ◽  
Soliton Solutions ◽  
Wave Interactions ◽  
N Wave ◽  
Explicit Expressions

We study matrix Darboux transformation (MDT) for the generalized N-wave interactions (local/nonlocal) and express multi-soliton solutions in terms of ratios of determinants. In order to explain our construction, we obtain explicit expressions of single and double soliton solutions for local/nonlocal 3-wave and 4-wave interactions.

Multi-component noncommutative coupled dispersionless system and its quasideterminant solutions

2018 ◽  
Vol 33 (15) ◽  
pp. 1850086 ◽  
Author(s):  
H. Wajahat A. Riaz ◽  
Mahmood ul Hassan
Keyword(s):  
Darboux Transformation ◽  
Lax Pair ◽  
Zero Curvature Representation ◽  
Zero Curvature ◽  
Multisoliton Solutions ◽  
Explicit Expressions

The multi-component noncommutative coupled dispersionless (NC-CD) system is presented. It has been shown that multi-component NC-CD system is integrable in the sense of exhibiting its Lax pair, zero-curvature representation, Darboux transformation and multisoliton solutions. Explicit expressions of multisoliton solutions of this noncommutative system have been computed and results have been compared with their commutative counterparts.

scholarly journals Non-local matrix elements in B(s) → {K(*), ϕ}ℓ+ℓ−

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Nico Gubernari ◽  
Danny van Dyk ◽  
Javier Virto
Keyword(s):  
Light Cone ◽  
Truncation Error ◽  
Sum Rules ◽  
Form Factors ◽  
Numerical Results ◽  
Matrix Elements ◽  
Current State ◽  
Theoretical Predictions ◽  
The Matrix ◽  
Non Local

Abstract We revisit the theoretical predictions and the parametrization of non-local matrix elements in rare $$ {\overline{B}}_{(s)}\to \left\{{\overline{K}}^{\left(\ast \right)},\phi \right\}{\mathrm{\ell}}^{+}{\mathrm{\ell}}^{-} $$ B ¯ s → K ¯ ∗ ϕ ℓ + ℓ − and $$ {\overline{B}}_{(s)}\to \left\{{\overline{K}}^{\ast },\phi \right\}\gamma $$ B ¯ s → K ¯ ∗ ϕ γ decays. We improve upon the current state of these matrix elements in two ways. First, we recalculate the hadronic matrix elements needed at subleading power in the light-cone OPE using B-meson light-cone sum rules. Our analytical results supersede those in the literature. We discuss the origin of our improvements and provide numerical results for the processes under consideration. Second, we derive the first dispersive bound on the non-local matrix elements. It provides a parametric handle on the truncation error in extrapolations of the matrix elements to large timelike momentum transfer using the z expansion. We illustrate the power of the dispersive bound at the hand of a simple phenomenological application. As a side result of our work, we also provide numerical results for the Bs → ϕ form factors from B-meson light-cone sum rules.

Unified approach to the representations of groups SU(2) and SU(1, 1)

1970 ◽  
Vol 48 (10) ◽  
pp. 1272-1282
Author(s):  
Klang-Chuen Young
Keyword(s):  
Generalized Functions ◽  
Irreducible Representations ◽  
Matrix Elements ◽  
Unified Approach ◽  
Canonical Bases ◽  
Representations Of Groups ◽  
Special Cases ◽  
The Matrix ◽  
Finite Transformation ◽  
Explicit Expressions

A unified approach to the representations of groups SU(2) and SU(1, 1) is made. The method is based on the observation that SU(2) and SU(1, 1) can be considered as special cases of a group G(a). The representation of G(a) is realized in the space of homogeneous generalized functions. The canonical bases of the unitary irreducible representations are constructed explicitly. The matrix elements for the finite transformation are found. Explicit expressions for the Wigner coefficients are also obtained.

scholarly journals DARBOUX TRANSFORMATION AND MULTI-SOLITON SOLUTIONS OF TWO-BOSON HIERARCHY

2011 ◽  
Vol 26 (09) ◽  
pp. 625-636 ◽  
Author(s):  
ASHOK DAS ◽  
U. SALEEM
Keyword(s):  
Linear Equation ◽  
Darboux Transformation ◽  
Soliton Solutions ◽  
Integrable Models ◽  
Darboux Transformations ◽  
The Matrix

We study Darboux transformations for the two boson (TB) hierarchy both in the scalar as well as in the matrix descriptions of the linear equation. While Darboux transformations have been extensively studied for integrable models based on SL (2, R) within the AKNS framework, this model is based on SL (2, R)⊗ U (1). The connection between the scalar and the matrix descriptions in this case implies that the generic Darboux matrix for the TB hierarchy has a different structure from that in the models based on SL (2, R) studied thus far. The conventional Darboux transformation is shown to be quite restricted in this model. We construct a modified Darboux transformation which has a much richer structure and which also allows for multi-soliton solutions to be written in terms of Wronskians. Using the modified Darboux transformations, we explicitly construct one-soliton/kink solutions for the model.

scholarly journals Non-local homogenized limits for composite media with highly anisotropic periodic fibres

2006 ◽  
Vol 136 (1) ◽  
pp. 87-114 ◽  
Author(s):  
K. D. Cherednichenko ◽  
V. P. Smyshlyaev ◽  
V. V. Zhikov
Keyword(s):  
Type Inequality ◽  
Double Porosity ◽  
High Contrast ◽  
Composite Media ◽  
Homogenization Problem ◽  
The Matrix ◽  
Non Local ◽  
The Green Function ◽  
Anisotropic Conducting ◽  
Scale Limit

We consider a homogenization problem for highly anisotropic conducting fibres embedded into an isotropic matrix. For a ‘double porosity’-type scaling in the expression of high contrast between the conductivity along the fibres and the conductivities in the transverse directions, we prove the homogenization theorem and derive two-scale homogenized equations using a version of the method of two-scale convergence, supplemented in the case when the spectral parameter λ = 0 by a newly derived variant of high-contrast Poincaré-type inequality. Further elimination of the 'rapid' component from the two-scale limit equations results in a non-local (convolution-type integro-differential) equation for the slowly varying part in the matrix, with the non-local kernel explicitly related to the Green function on the fibre. The regularity of the solution to the non-local homogenized equation is proved.

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