Under the constrained condition induced by the eigenfunction expresson of the potential (u, v)T = (-[A2q, q], [A2p, p])T = f (q, p), the spatial part of the Lax pair of the Kaup-Newell equation is non linearized to be...Under the constrained condition induced by the eigenfunction expresson of the potential (u, v)T = (-[A2q, q], [A2p, p])T = f (q, p), the spatial part of the Lax pair of the Kaup-Newell equation is non linearized to be a completely integrable system (R2N, Adp AND dq, H = H-1) with the Hamiltonian H-1 = -[A3q, p]-1/2[A2p, p][A2q, q]. while the nonlinearization of the time part leads to its N-involutive system {H(m)}. The involutive solution of the compatible fsystem (H-1), (H(m)) is mapped by into the solution of the higher order Kaup-Newell equation.展开更多
A variant of Fermat’s last Diophantine equation is proposed by adjusting the number of terms in accord with the power of terms and a theorem describing the solubility conditions is stated. Numerically obtained primit...A variant of Fermat’s last Diophantine equation is proposed by adjusting the number of terms in accord with the power of terms and a theorem describing the solubility conditions is stated. Numerically obtained primitive solutions are presented for several cases with number of terms equal to or greater than powers. Further, geometric representations of solutions for the second and third power equations are devised by recasting the general equation in a form with rational solutions less than unity. Finally, it is suggested to consider negative and complex integers in seeking solutions to Diophantine forms in general.展开更多
The aim of this work is to find exact solutions of the Dirac equation in(1+1) space-time beyond the already known class.We consider exact spin(and pseudo-spin) symmetric Dirac equations where the scalar potential is e...The aim of this work is to find exact solutions of the Dirac equation in(1+1) space-time beyond the already known class.We consider exact spin(and pseudo-spin) symmetric Dirac equations where the scalar potential is equal to plus(and minus) the vector potential.We also include pseudo-scalar potentials in the interaction.The spinor wavefunction is written as a bounded sum in a complete set of square integrable basis,which is chosen such that the matrix representation of the Dirac wave operator is tridiagonal and symmetric.This makes the matrix wave equation a symmetric three-term recursion relation for the expansion coefficients of the wavefunction.We solve the recursion relation exactly in terms of orthogonal polynomials and obtain the state functions and corresponding relativistic energy spectrum and phase shift.展开更多
文摘Under the constrained condition induced by the eigenfunction expresson of the potential (u, v)T = (-[A2q, q], [A2p, p])T = f (q, p), the spatial part of the Lax pair of the Kaup-Newell equation is non linearized to be a completely integrable system (R2N, Adp AND dq, H = H-1) with the Hamiltonian H-1 = -[A3q, p]-1/2[A2p, p][A2q, q]. while the nonlinearization of the time part leads to its N-involutive system {H(m)}. The involutive solution of the compatible fsystem (H-1), (H(m)) is mapped by into the solution of the higher order Kaup-Newell equation.
文摘A variant of Fermat’s last Diophantine equation is proposed by adjusting the number of terms in accord with the power of terms and a theorem describing the solubility conditions is stated. Numerically obtained primitive solutions are presented for several cases with number of terms equal to or greater than powers. Further, geometric representations of solutions for the second and third power equations are devised by recasting the general equation in a form with rational solutions less than unity. Finally, it is suggested to consider negative and complex integers in seeking solutions to Diophantine forms in general.
基金King Fahd University of Petroleum and Minerals (KFUPM) for their support under research grant RG1502the material support and encouragements of the Saudi Center for Theoretical Physics (SCTP)
文摘The aim of this work is to find exact solutions of the Dirac equation in(1+1) space-time beyond the already known class.We consider exact spin(and pseudo-spin) symmetric Dirac equations where the scalar potential is equal to plus(and minus) the vector potential.We also include pseudo-scalar potentials in the interaction.The spinor wavefunction is written as a bounded sum in a complete set of square integrable basis,which is chosen such that the matrix representation of the Dirac wave operator is tridiagonal and symmetric.This makes the matrix wave equation a symmetric three-term recursion relation for the expansion coefficients of the wavefunction.We solve the recursion relation exactly in terms of orthogonal polynomials and obtain the state functions and corresponding relativistic energy spectrum and phase shift.