On Lorentz-invariant 2D equations admitting long-lived localized solutions with a nontrivial structure
aa r X i v : . [ n li n . PS ] S e p sent to Jetp Letters On Lorentz-invariant 2D equations admitting long-lived localized solutions with anontrivial structure.
R.K. Salimov , T.R. Salimov , E.G. Ekomasov Bashkir State University, Ufa, Russia Moscow Institute of Physics and Tehnology, Dolgoprudny, Russiae-mail: [email protected]
Abstract
The article studies Lorentz-invariant 2D equations with long-lived ( t ∽ Keywords : nonlinear differential equations, soliton, confinement model by flux tube.
Introduction
Soliton solutions of nonlinear equations are often regarded as extended models of particles[1-9], for example, the Skyrme model [10-12] describing the internal structure of baryons andlightweight nuclei. In the course of this approach particles are described as a “bunched field”occupying a bounded region of a space. Such an approach is different from the standard modelwhere the extended particles are constructed of point particles. One of the drawbacks of thesoliton approach is a small number of mathematical models with 2D and 3D stable localizedsolutions. The work presents Lorentz-invariant 2D equations with long-lived localized solutions.The solutions have an interesting internal structure the properties of which suggest the analogywith the hadrons’ structure.
Results
Equations (1,2) with localized symmetric solutions that do not spread spherically or cylin-drically in 3D and 2D cases have been considered before in [13]. u rr + 2 u r r − u tt = u k +1 (1) u rr + u r r − u tt = u k +1 (2)Numerical simulation in 2D case shows that without the condition of cylindrical symmetrythe localized solutions of one field are unstable and are quite fast ( t ∽
30) to fall apart. Forthis reason equations of the form (3-4) for two fields were considered u xx + u yy − u tt = α u ( u + v ) n n +1 + βu + γu ( u + v ) (3) xx + v yy − v tt = α v ( u + v ) n n +1 + βv + γv ( u + v ) (4)The equations are interesting because they have long-lived nonspreading numerical solu-tions. Localized solutions in them are kept without noticable energy losses during the wholetime of the numerical simulation ( t ∽ Figure 1: Orientation of the solution (3-4) in Ox-direction. n = 8; α = 1; β = 12; γ = 12Figure 2: Orientation of the solution in Oy-direction. To obtain a more complex internal structure, solutions of equations (5-7) for three scalarfields were numerically investigated: u xx + u yy − u tt = α u ( u + v + w ) n n +1 + βu + γu ( u + v + w ) + λuv + ξw (5) v xx + v yy − v tt = α v ( u + v + w ) n n +1 + βv + γv ( u + v + w ) + λvu − ξw (6) w xx + w yy − w tt = α w ( u + v + w ) n n +1 + ηw + γu ( u + v + w ) + ξ ( u − v )++ µ w ( w ) n n +1 (7) u, v repel at largeamplitudes because of summands . The presence of several fields, simultaneously nonzero,results in a more stable localization of solutions. The example of the existence of the solutionsof equations (3-4) proves that. Numerical solutions of equations (5-7) have a more remarkableinternal structure. Under some initial conditions and parameters ( n = 8; α = 1; β = 12; γ =18; λ = 360; ξ = 1 . η = 8 π ; µ = 1 ) , the solution is a structure of two periodically emergingmaxima of the value of function ( u + v + w ) (1 / ) . See fig. 3. Figure 3:
The solution with a maximal amplitude is changed to the solution with a minimal ampli-tude of value ( u + v + w ) (1 / . See fig. 4 Figure 4:Figure 5:
Such a behavior of the solutions is kept constant for quite a long time ( t ∽ esults and discussion As is seen from the numerical solution of equations (3-4) the equations of the scalar fields withfractional nonlinear nature are of interest for being the equations with nonspreading localizedsolutions. The existence of deviation of spatial orientation of the solutions suggests some analogof surface tension for localized solutions. Repelling for different fields can be compensated bythis surface tension. The fact makes it possible to obtain long-lived solutions with severalspatial maxima, which is proved by the numerical solutions of equations (5-7). The solutionsof equations (5-7) can be interpreted as the classical variant of the model consisting of two“quarks”, or spatially separated maxima of fields. In this case, the solutions of the equationsare similar to the confinement model by flux tube [14]. Besides, model (3-4) is promising becausein the Hamiltonian of the model the summand of the form vu providing some confinement forfields can be added. This is the case when, if one or more than one field u, v is different fromzero, the summand generates another field in motion equations (3-4). References [1] D. J. Kaup, A. C. Newell, Solitons as particles, oscillators, and in slowly changing media:a singular perturbation theory, R. Soc. Lond. A361413–446[2] A. M. Kosevich, Particle and wave properties of solitons: Resonant and non-resonantsoliton scattering by impurities, Physica D, V. 41,I. 2, (1990).[3] E. Jenkins, A. V. Manohar, M. B. Wise, M B. Baryons containing a heavy quark as solitons.Netherlands: N. p., 1993. Web. doi:10.1016/0550-3213(93)90256-O.[4] Yu. P. Rybakov, B. Saha, Physics Letters A 122, 5 (1996)[5] N. S. Manton, Solitons as elementary particles: a paradigm scrutinized. , Nonlinearity,21(11):T221, (2008)[6] A. Maccari, Nonlinear Field Equations and Solitons as Particles, EJTP 3, No. 10 (2006)39–88[7] E. J. Weinberg . Classical Solutions in Quantum Field Theory. Cambridge University Press(2012).[8] M. J. Ablowitz, Nonlinear Dispersive Waves: Asymptotic Analysis and Solitons. Cam-bridge University Press (2011)[9] Y. Iwata, Solitons in Nuclear Time-Dependent Density Functional Theory, Front. Phys.,30 June 2020.[10] C. Adam, C. Naya, J. Sanchez-Guillen, A. Wereszczynski, Physical review letters, 111(23), 232501[11] C. Naya, P. Sutcliffe, Physical Review Letters 121 (23), 232002[12] R. A. Battye, N. S. Manton, P. M. Sutcliffe, Skyrmions and the α -particle model of nuclei,Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.463, issue 2077, pp. 261-279[13] E. G. Ekomasov, R. K. Salimov, Jetp Lett. 100, 477–480 (2014).[14] K. G. Wilson, Physical Review D. 10 (8): 2445–2459, (1974).-particle model of nuclei,Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.463, issue 2077, pp. 261-279[13] E. G. Ekomasov, R. K. Salimov, Jetp Lett. 100, 477–480 (2014).[14] K. G. Wilson, Physical Review D. 10 (8): 2445–2459, (1974).