Riemann–Hilbert problem for the defocusing Lakshmanan–Porsezian–Daniel equation with fully asymmetric nonzero boundary conditions

The Riemann–Hilbert approach is demonstrated to investigate the defocusing Lakshmanan–Porsezian–Daniel equation under fully asymmetric nonzero boundary conditions.In contrast to the symmetry case,this paper focuses on the branch points related to the scattering problem rather than using the Riemann surfaces.For the direct problem,we analyze the Jost solution of lax pairs and some properties of scattering matrix,including two kinds of symmetries.The inverse problem at branch points can be presented,corresponding to the associated Riemann–Hilbert.Moreover,we investigate the time evolution problem and estimate the value of solving the solutions by Jost function.For the inverse problem,we construct it as a Riemann–Hilbert problem and formulate the reconstruction formula for the defocusing Lakshmanan–Porsezian–Daniel equation.The solutions of the Riemann–Hilbert problem can be constructed by estimating the solutions.Finally,we work out the solutions under fully asymmetric nonzero boundary conditions precisely via utilizing the Sokhotski–Plemelj formula and the square of the negative column transformation with the assistance of Riemann surfaces.These results are valuable for understanding physical phenomena and developing further applications of optical problems.

Numerical simulation for the comparison of thermal and plastic material flow behavior between symmetrical and asymmetrical boundary conditions during friction stir welding

An accurate description of the contact condition between the tool and the workpiece material is one of the most important issues for expounding the underlying multi-physics coupled mechanism during friction stir welding(FSW)process.In the present study,a novel asymmetrical boundary condition around the tool-workpiece contact interface is proposed for the FSW of AA2024-T4 alloy.A three-dimensional computational fluid dynamics model is employed for the comparison of the coupled thermal and plastic material flow behavior between asymmetrical and symmetrical boundary conditions.Numerical results of heat generation,temperature distribution,tunnel defect formation and material flow streamline during the welding process are quantitatively analyzed.Besides,various experimental measuring methods are utilized to obtain information about temperature,thermal cycle,tool torque and horizontal cross-section around the exiting keyhole.It is revealed that the modeling results of heat flux density and temperature distribution around the pin,as well as material flow characteristics all change significantly for the two models with different boundary conditions.Particularly,the asymmetrical boundary condition is more capable of predicting temperature fluctuation,plastic material flow along the vertical direction,as well as tunnel defect formation during FSW.Therefore,the superiority of the model with asymmetrical boundary condition over the symmetrical one during the numerical simulation of FSW is elucidated.