Higher order Darboux transformations in two dimensions, linearizable auxiliary equations, and invariant potentials

2011 ◽  
Vol 52 (8) ◽  
pp. 083505 ◽  
Author(s):  
Axel Schulze-Halberg
Keyword(s):  
Higher Order ◽  
Two Dimensions ◽  
Darboux Transformations

scholarly journals Higher-order Darboux transformations for two-dimensional Dirac systems with diagonal matrix potential

2021 ◽  
Vol 2090 (1) ◽  
pp. 012038
Author(s):  
A Schulze-Halberg
Keyword(s):  
Dirac Equation ◽  
Explicit Form ◽  
Higher Order ◽  
Diagonal Matrix ◽  
Two Dimensional ◽  
Darboux Transformations ◽  
Dirac Systems ◽  
Matrix Potential ◽  
De Vries ◽  
The Matrix

Abstract We construct the explicit form of higher-order Darboux transformations for the two-dimensional Dirac equation with diagonal matrix potential. The matrix potential entries can depend arbitrarily on the two variables. Our construction is based on results for coupled Korteweg-de Vries equations [27].

scholarly journals Nonlinear higher-order spectral solution for a two-dimensional moving load on ice

2009 ◽  
Vol 621 ◽  
pp. 215-242 ◽  
Author(s):  
FÉLICIEN BONNEFOY ◽  
MICHAEL H. MEYLAN ◽  
PIERRE FERRANT
Keyword(s):  
Critical Speed ◽  
Moving Boundary ◽  
Moving Load ◽  
Phase Speed ◽  
Nonlinear Effects ◽  
Higher Order ◽  
Two Dimensions ◽  
Linear Solution ◽  
Nonlinear Solution ◽  
Bernoulli’S Equation

We calculate the nonlinear response of an infinite ice sheet to a moving load in the time domain in two dimensions, using a higher-order spectral method. The nonlinearity is due to the moving boundary, as well as the nonlinear term in Bernoulli's equation and the elastic plate equation. We compare the nonlinear solution with the linear solution and with the nonlinear solution found by Parau & Dias (J. Fluid Mech., vol. 460, 2002, pp. 281–305). We find good agreement with both solutions (with the correction of an error in the Parau & Dias 2002 results) in the appropriate regimes. We also derive a solitary wavelike expression for the linear solution – close to but below the critical speed at which the phase speed has a minimum. Our model is carefully validated and used to investigate nonlinear effects. We focus in detail on the solution at a critical speed at which the linear response is infinite, and we show that the nonlinear solution remains bounded. We also establish that the inclusion of nonlinearities leads to significant new behaviour, which is not observed in the linear solution.

Noncommutative coupled complex modified Korteweg–de Vries equation: Darboux and binary Darboux transformations

2019 ◽  
Vol 34 (07n08) ◽  
pp. 1950054
Author(s):  
H. Wajahat A. Riaz
Keyword(s):  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Vries Equation ◽  
Soliton Solutions ◽  
Higher Order ◽  
Nonlinear Evolution Equations ◽  
Order Effects ◽  
Darboux Transformations ◽  
De Vries ◽  
Higher Order Effects

Higher-order nonlinear evolution equations are important for describing the wave propagation of second- and higher-order number of fields in optical fiber systems with higher-order effects. One of these equations is the coupled complex modified Korteweg–de Vries (ccmKdV) equation. In this paper, we study noncommutative (nc) generalization of ccmKdV equation. We present Darboux and binary Darboux transformations (DTs) for the nc-ccmKdV equation and then construct its Quasi-Grammian solutions. Further, single and double-hump soliton solutions of first- and second-order are given in commutative settings.

scholarly journals The Performance of the Higher-Order Radiation Condition Method for the Penetrable Cylinder

2007 ◽  
Vol 2007 ◽  
pp. 1-14 ◽  
Author(s):  
Bülent Yilmaz
Keyword(s):  
Total Pressure ◽  
Wave Speed ◽  
Surrounding Medium ◽  
Radiation Condition ◽  
Higher Order ◽  
Two Dimensions ◽  
Total Field ◽  
Condition Method ◽  
Cartesian System ◽  
Total Velocity

We have considered the scattering of a plane wave by a penetrable acoustic circular cylinder. The boundary conditions are continuity of the total pressure and the total velocity. The wave speed and density of the target are different from that of the surrounding medium. We investigated the performance of higher-order SRCs up toL4operator in two dimensions. We assume that in the rectangular Cartesian system of axes,(x,y,z), thezaxis coincides with the axis of the cylinder and an incident wave propagates in a direction perpendicular to thezaxis. All the field quantities are then independent ofz. Numerical results are added to present the change of the module of the total field and the magnitude of the far field with respect toθ.

HIERARCHICAL DIMENSIONS OF COPING STYLES: A STUDY CONDUCTED WITH TURKISH UNIVERSITY STUDENTS

2006 ◽  
Vol 34 (5) ◽  
pp. 525-534 ◽  
Author(s):  
Faruk Gençöz ◽  
Tülin Gençöz ◽  
Özlem Bozo
Keyword(s):  
Factor Analysis ◽  
University Students ◽  
Coping Styles ◽  
Reliability And Validity ◽  
Higher Order ◽  
Two Dimensions ◽  
Good Reliability ◽  
Empirical Examination ◽  
Turkish University Students ◽  
Third Dimension

An empirical examination was carried out of the hierarchical dimensions of coping styles in a Turkish sample. Ways of coping data were collected from 194 university students, subjected to factor analysis and 5 factors were obtained. These factors were subjected to second-order factor analysis which revealed 3 main dimensions. As predicted the first two dimensions were problem-focused coping and emotion-focused coping. The third dimension consisted of items related to seeking social support, implying the presence and importance of an indirect coping style. In addition to construct validity, Guttman split-half reliability and criterion validity of these three higher order factors revealed good reliability and validity outcomes. It was also emphasized that these 3 higher order factors constituted independent dimensions of coping styles. Implications of the results are discussed.

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