Conservation laws of partial differential equations: Symmetry, adjoint symmetry and nonlinear self-adjointness

Nonlinear self-adjointness method for constructing conservation laws of partial differential equations (PDEs) is further studied. We show that any adjoint symmetry of PDEs is a differential substitution of nonlinear self-adjointness and vice versa. Consequently, each symmetry of PDEs corresponds to a conservation law via a formula if the system of PDEs is nonlinearly self-adjoint with differential substitution. As a byproduct, we find that the set of differential substitutions includes the set of conservation law multipliers as a subset. The results are illustrated by three typical examples

An upper order bound of the invariant manifold in Lax pairs of a nonlinear evolution partial differential equation

In \cite{hab-2016,hab-2017}, Habibullin \emph{et.al} proposed an approach to construct Lax pairs of a nonlinear integrable partial differential equation (PDE), where one is the linearized equation of the studied PDE and the other is the invariant manifold of the linearized equation. In this paper, we show that the invariant manifold is the characteristic of a generalized conditional symmetry of the system composed of the studied PDE and its linearized PDE. Then we give an upper order bound of the invariant manifold which provides a theoretical basis for a complete classification of such type of invariant manifold. Moreover, we suggest a modified method to construct Lax pair of the KdV equation which can not be obtained by the original method in \cite{hab-2016,hab-2017}

The (1,0)+(0,1) spinor description of the photon field and its preliminary applications

Because spatio-temporal tensors are associated with the Lorentz group, whereas spinors are associated with its covering group SL(2, C), one can associate with every tensor a spinor (but not vice versa). In particular, the (1,0)+(0,1) representation of SL(2, C) can provide a six-component spinor equivalent to the electromagnetic field tensor. The chief aim of this work is to develop the (1,0)+(0,1) description for the electromagnetic field in the absence of sources, rigorously and systematically, which should be useful if we are to deal with those issues involving with single-photon states and the angular momentum of light, etc. Based on our formalism, the quantum theory and some symmetries of the photon field can be discussed in a new manner, and the spin-orbit interaction of photons can be described in a form that is closely analogous to that of the Dirac electron. Moreover, in terms of the (1,0)+(0,1) description, one can treat the photon field in curved spacetime via spin connection and the tetrad formalism, which is of great advantage to study the gravitational spin-orbit coupling of photons.Comment: 60 pages, no figure. It will provide a comprehensive reference for some of our future work

Reconsideration of photonic tunneling through undersized waveguides

All the previous studies on photonic tunneling are just based on a simple and directly analogy with a one-dimensional quantum-mechanical tunneling, without taking into account the horizontal structure of electromagnetic waves along the waveguide, such that they are oversimplified and incomplete. Here we present a more serious deliberation on photonic tunneling through cut-off waveguides, and obtain a strictly theoretical model with some new results.Comment: 22 pages, no figur

Approximate homotopy series solutions of perturbed PDEs via approximate symmetry method

We show that the two couple equations derived by approximate symmetry method and approximate homotopy symmetry method are connected by a transformation for the perturbed PDEs. Consequently, approximate homotopy series solutions can be obtained by acting the transformation on the known solutions by approximate symmetry method. Applications to the Cahn-Hilliard equation illustrate the effectiveness of the transformation

Does the bottomonium counterpart of X(3872)X(3872) exist?

A narrow line shape peak at about 10615 MeV, just above the threshold in the $B\bar B^*$ channel, which can be regarded as the signal of bottomonium counterpart of $X(3872)$, $X_b$, is predicted by using the extended Friedrichs scheme. Though a virtual state is found at about 10593 MeV in this scheme, we point out that the peak is contributed mainly by the coupling form factor, which comes from the convolution of the interaction term and meson wave functions including the one from $\chi_{b1}(4P)$, but not mainly by the virtual-state pole. In this picture, the reason why $X_b$ signal is not observed in the $\Upsilon\pi^+\pi^-$ and $\Upsilon\pi^+\pi^-\pi^0$ channels can also be understood. The $\chi_{b1}(4P)$ mass and width are found to be about 10771 MeV and 6 MeV, respectively and a dynamically generated broad resonance is also found with its mass and width at about 10672 MeV and 78 MeV, respectively. The line shapes of these two states are also affected by the form factor effect. Thus, this study also emphasizes the importance of the structure of the wave functions of high radial excitations in the analysis of the line shapes, and provides a caveat that some signals may be generated from the structures of the form factors rather than from poles.Comment: 5 pages, 3 figures; v2, the final published versio

Comprehending Isospin breaking effects of X(3872)X(3872) in a Friedrichs-model-like scheme

Recently, we have shown that the $X(3872)$ state can be naturally generated as a bound state by incorporating the hadron interactions into the Godfrey-Isgur quark model using the Friedrichs-like model combined with the QPC model, in which the wave function for the $X(3872)$ as a combination of the bare $c\bar c$ state and the continuum states can also be obtained. Under this scheme, we now investigate the isospin breaking effect of $X(3872)$ in its decays to $J/\psi\pi^+\pi^-$ and $J/\psi\pi^+\pi^-\pi^0$. By Considering its dominant continuum parts coupling to $J/\psi\rho$ and $J/\psi\omega$ through the quark rearrangement process, one could obtain the reasonable ratio of $\mathcal{B}(X(3872)\rightarrow J/\psi\pi^+\pi^-\pi^0)/\mathcal{B}(X(3872)\rightarrow J/\psi\pi^+\pi^-)\simeq (0.58\sim 0.92)$. It is also shown that the $\bar D D^*$ invariant mass distributions in the $B\rightarrow \bar D D^* K$ decays could be understood qualitatively at the same time. This scheme may provide more insight to understand the enigmatic nature of the $X(3872)$ state.Comment: 13 pages, 4 figure

The origin of light 0+0^{+} scalar resonances

We demonstrate how most of the light $J^{P}=0^{+}$ spectrum below $2.0\,\mathrm{GeV}$ and their decays can be consistently described by the unitarized quark model incorporating the chiral constraints of Adler zeros and taking SU(3) breaking effects into account. These resonances appear as poles in the complex $s$ plane in a unified picture as $q\bar{q}$ states strongly dressed by hadron loops. Through the large $N_c$ analysis, these resonances are found to naturally separate into two kinds: $\sigma, \kappa, f_0(980), a_0(980)$ are dynamically generated and run away from the real axis as $N_c$ increases, while the others move towards the $q\bar q$ seeds. In this picture, the line shape of $a_0(980)$ is produced by a broad pole below the $K\bar{K}$ threshold, and exhibits characteristics similar to the $\sigma$ and $\kappa$.Comment: 7 pages, 12 figures, Revtex4-1. Significantly revised and expanded version. The main result not change

Understanding X(3862)X(3862), X(3872)X(3872), and X(3930)X(3930) in a Friedrichs-model-like scheme

We developed a Friedrichs-model-like scheme in studying the hadron resonance phenomenology and present that the hadron resonances might be regarded as the Gamow states produced by a Hamiltonian in which the bare discrete state is described by the result of usual quark potential model and the interaction part is described by the quark pair creation model. In a one-parameter calculation, the $X(3862)$, $X(3872)$, and $X(3930)$ state could be simultaneously produced with a quite good accuracy by coupling the three P-wave states, $\chi_{c2}(2P)$, $\chi_{c1}(2P)$, $\chi_{c0}(2P)$ predicted in the Godfrey-Isgur model to the $D\bar D$, $D\bar D^{*}$, $D^*\bar D^*$ continuum states. At the same time, we predict that the $h_c(2P)$ state is at about 3902 MeV with a pole width of about 54 MeV. In this calculation, the $X(3872)$ state has a large compositeness. This scheme may shed more light on the long-standing problem about the general discrepancy between the prediction of the quark model and the observed values, and it may also provide reference for future search for the hadron resonance state.Comment: 5 pages, 1 figure; A mistake was found in the numerical calculation and the numerical results change a little. The qualitative discussion and conclusion not change

New interpretation to zitterbewegung

In previous investigations on zitterbewegung(zbw) of electron, it is believed that the zbw results from some internal motion of electron. However, all the analyses are made at relativistic quantum mechanical level. In framework of quantum field theory (QFT), we find that the origin of zbw is different from previous conclusion. Especially, some new interesting conclusions are derived at this level: 1) the zbw arises from the rapid to-and-fro polarization of the vacuum in the range of the Compton wavelength (divided by $4\pi$) of the electron, which offer the four-dimensional(4D) spin and intrinsic electromagnetic-moment tensor to the electron; 2) Any attempt that attributes spin (rather than double the spin) of the electron to some kind of orbital angular momentum would not be successful; 3) the macroscopic classical speed of the Dirac vacuum medium vanish in all inertial systems.Comment: 5 pages, no figures, revte