Regular and Singular Pulse and Front Solutions and Possible Isochronous Behavior in the Short-Pulse Equation: Phase-Plane, Multi-Infinite Series and Variational Approaches

In this paper we employ three recent analytical approaches to investigate the possible classes of traveling wave solutions of some members of a family of so-called short-pulse equations (SPE). A recent, novel application of phase-plane analysis is first employed to show the existence of breaking kink wave solutions in certain parameter regimes. Secondly, smooth traveling waves are derived using a recent technique to derive convergent multi-infinite series solutions for the homoclinic (heteroclinic) orbits of the traveling-wave equations for the SPE equation, as well as for its generalized version with arbitrary coefficients. These correspond to pulse (kink or shock) solutions respectively of the original PDEs. Unlike the majority of unaccelerated convergent series, high accuracy is attained with relatively few terms. And finally, variational methods are employed to generate families of both regular and embedded solitary wave solutions for the SPE PDE. The technique for obtaining the embedded solitons incorporates several recent generalizations of the usual variational technique and it is thus topical in itself. One unusual feature of the solitary waves derived here is that we are able to obtain them in analytical form (within the assumed ansatz for the trial functions). Thus, a direct error analysis is performed, showing the accuracy of the resulting solitary waves. Given the importance of solitary wave solutions in wave dynamics and information propagation in nonlinear PDEs, as well as the fact that not much is known about solutions of the family of generalized SPE equations considered here, the results obtained are both new and timely.Comment: accepted for publication in Communications in Nonlinear Science and Numerical Simulatio

Deciphering the regulatory landscapte of fetal and adult γδ T-cell development at single-cell resolution

γδ T cells with distinct properties develop in the embryonic and adult thymus and have been identified as critical players in a broad range of infections, antitumor surveillance, autoimmune diseases, and tissue homeostasis. Despite their potential value for immunotherapy, differentiation of γδ T cells in the thymus is incompletely understood. Here, we establish a high‐resolution map of γδ T‐cell differentiation from the fetal and adult thymus using single‐cell RNA sequencing. We reveal novel sub‐types of immature and mature γδ T cells and identify an unpolarized thymic population which is expanded in the blood and lymph nodes. Our detailed comparative analysis reveals remarkable similarities between the gene networks active during fetal and adult γδ T‐cell differentiation. By performing a combined single‐cell analysis of Sox13, Maf, and Rorc knockout mice, we demonstrate sequential activation of these factors during IL ‐17‐producing γδ T‐cell (γδT17) differentiation. These findings substantially expand our understanding of γδ T‐cell ontogeny in fetal and adult life. Our experimental and computational strategy provides a blueprint for comparing immune cell differentiation across developmental stages

Non-classical symmetries and the singular manifold method: A further two examples

This paper discusses two equations with the conditional Painleve property. The usefulness of the singular manifold method as a tool for determining the non-classical symmetries that reduce the equations to ordinary differential equations with the Painleve property is confirmed once moreComment: 9 pages (latex), to appear in Journal of Physics

Non-classical symmetries and the singular manifold method revisited

The connection between the singular manifold method (Painlevé expansions truncated at the constant term) and symmetry reductions of two members of a family of Cahn-Hilliard equations is considered. The conjecture that similarity information for a nonlinear partial differential equation may always be fully recovered from the singular manifold method is violated for these equations, and is thus shown to be invalid in general. Given that several earlier examples demonstrate the connection between the two techniques in some cases, it now becomes necessary to establish when such a relationship exists - a question related to a deeper understanding of Painlevé analysis. This issue is also briefly discussed

Regular and singular pulse and front solutions and possible isochronous behavior in the Extended-Reduced Ostrovsky Equation: Phase-plane, multi-infinite series and variational formulations

In this paper we employ three recent analytical approaches to investigate several classes of traveling wave solutions of the so-called extended-reduced Ostrovsky Equation (exROE). A recent extension of phase-plane analysis is first employed to show the existence of breaking kink wave solutions and smooth periodic wave (compacton) solutions. Next, smooth traveling waves are derived using a recent technique to derive convergent multi-infinite series solutions for the homoclinic orbits of the traveling-wave equations for the exROE equation. These correspond to pulse solutions respectively of the original PDEs. We perform many numerical tests in different parameter regime to pinpoint real saddle equilibrium points of the corresponding traveling-wave equations, as well as ensure simultaneous convergence and continuity of the multi-infinite series solutions for the homoclinic orbits anchored by these saddle points. Unlike the majority of unaccelerated convergent series, high accuracy is attained with relatively few terms. And finally, variational methods are employed to generate families of both regular and embedded solitary wave solutions for the exROE PDE. The technique for obtaining the embedded solitons incorporates several recent generalizations of the usual variational technique and it is thus topical in itself. One unusual feature of the solitary waves derived here is that we are able to obtain them in analytical form (within the assumed ansatz for the trial functions). Thus, a direct error analysis is performed, showing the accuracy of the resulting solitary waves. Given the importance of solitary wave solutions in wave dynamics and information propagation in nonlinear PDEs, as well as the fact that not much is known about solutions of the family of generalized exROE equations considered here, the results obtained are both new and timel

Lagrangian Dynamics And Possible Isochronous Behavior In Several Classes Of Non-Linear Second Order Oscillators Via The Use Of Jacobi Last Multiplier

Abstract In this paper, we employ the technique of Jacobi Last Multiplier (JLM) to derive Lagrangians for several important and topical classes of non-linear second-order oscillators, including systems with variable and parametric dissipation, a generalized anharmonic oscillator, and a generalized Lane-Emden equation. For several of these systems, it is very difficult to obtain the Lagrangians directly, i.e., by solving the inverse problem of matching the Euler-Lagrange equations to the actual oscillator equation. In order to facilitate the derivation of exact solutions, and also investigate possible isochronous behavior in the analyzed systems, we next invoke some recent theoretical results and attempt to map the potential term to either the simple harmonic oscillator or the isotonic potential for specific values of the coefficient parameters of each non-linear oscillator. We find non-trivial parameter sets corresponding to isochronous dynamics in some of the considered systems, but none in others. Finally, the Lagrangians obtained here are coupled to Noether\u27s theorem, leading to non-trivial conservation laws for several of the oscillators

Lagrangian dynamics and possible isochronous behavior in several classes of non-linear second order oscillators via the use of Jacobi last multiplier

In this paper, we employ the technique of Jacobi Last Multiplier (JLM) to derive Lagrangians for several important and topical classes of non-linear second-order oscillators, including systems with variable and parametric dissipation, a generalized anharmonic oscillator, and a generalized Lane\u2013Emden equation. For several of these systems, it is very difficult to obtain the Lagrangians directly, i.e., by solving the inverse problem of matching the Euler\u2013Lagrange equations to the actual oscillator equation. In order to facilitate the derivation of exact solutions, and also investigate possible isochronous behavior in the analyzed systems, we next invoke some recent theoretical results and attempt to map the potential term to either the simple harmonic oscillator or the isotonic potential for specific values of the coefficient parameters of each non-linear oscillator. We find non-trivial parameter sets corresponding to isochronous dynamics in some of the considered systems, but none in others. Finally, the Lagrangians obtained here are coupled to Noether\u5f3s theorem, leading to non-trivial conservation laws for several of the oscillators

Phase Change Problem And Effects Of Geometrical Constants On Weld Pool Geometry In Sheet Metal Laser Welding

A three-dimensional quasi-state heat conduction model is presented for the solid/liquid interface in laser welding of thin sheet metals. The vapour plasma plume effect is incorporated into the model by considering the electromagnetic field equation to determine the laser power reaching the substrate surface after passing through the plume region. Optical properties, such as the dielectric constant and electrical conductivity of plume, and the thermal conductivity of the vapour of stainless steel 316 in the plume region are estimated. Elliptic-paraboloid weld pool geometry is considered and analytic expressions for temperature distributions are obtained in the solid and liquid regions based on the weld pool geometry. Experimental and theoretical results for the weld depths and widths are illustrated for process parameters such as the laser power, welding speed, and weld pool shape factors in x and y directions

Regular And Singular Pulse And Front Solutions And Possible Isochronous Behavior In The Extended-Reduced Ostrovsky Equation: Phase-Plane, Multi-Infinite Series And Variational Formulations

In this paper we employ three recent analytical approaches to investigate several classes of traveling wave solutions of the so-called extended-reduced Ostrovsky Equation (exROE). A recent extension of phase-plane analysis is first employed to show the existence of breaking kink wave solutions and smooth periodic wave (compacton) solutions. Next, smooth traveling waves are derived using a recent technique to derive convergent multi-infinite series solutions for the homoclinic orbits of the traveling-wave equations for the exROE equation. These correspond to pulse solutions respectively of the original PDEs. We perform many numerical tests in different parameter regime to pinpoint real saddle equilibrium points of the corresponding traveling-wave equations, as well as ensure simultaneous convergence and continuity of the multi-infinite series solutions for the homoclinic orbits anchored by these saddle points. Unlike the majority of unaccelerated convergent series, high accuracy is attained with relatively few terms. And finally, variational methods are employed to generate families of both regular and embedded solitary wave solutions for the exROE PDE. The technique for obtaining the embedded solitons incorporates several recent generalizations of the usual variational technique and it is thus topical in itself. One unusual feature of the solitary waves derived here is that we are able to obtain them in analytical form (within the assumed ansatz for the trial functions). Thus, a direct error analysis is performed, showing the accuracy of the resulting solitary waves. Given the importance of solitary wave solutions in wave dynamics and information propagation in nonlinear PDEs, as well as the fact that not much is known about solutions of the family of generalized exROE equations considered here, the results obtained are both new and timely

Effects Of Absorptivity, Shielding Gas Speed, And Contact Media On Sheet Metal Laser Welding

A three-dimensional quasi-steady state heat conduction model is developed for laser welding of sheet metals. The heat flux at the surface of the workpiece is considered to be due to a moving Gaussian laser beam. An analytical expression is obtained for the temperature distribution by solving the conduction problem using the Fourier integral transform technique. This expression is used to locate the melting temperature isotherm, and thereby determine the weld depth and width. Experimental and theoretical results for the weld depths and widths are illustrated for different welding parameters such as the laser power, absorptivity, welding speed, and shielding gas speed. The theory and experiment are found to agree reasonably well. The effects of absorptivity, shielding gas speed, and heat loss due to different contact media at the bottom surface of the workpiece are also investigated, and are found to be significant for thin metal laser welding