The travelling wave group is a stable wave packet. Many surprising results are derived from it. The group is easily quantized for photons and applied, as a solution to the relativistic Klein-Gordon equation, to free p...The travelling wave group is a stable wave packet. Many surprising results are derived from it. The group is easily quantized for photons and applied, as a solution to the relativistic Klein-Gordon equation, to free particles. Further solutions to the resulting algebraic equation provide a stable wave function for free antiparticles. Consistency with the superstructure of quantum electrodynamics is obtained by an assignment to the electron antiparticle of negative mass and negative charge. Then in 5-dimensional space-time-mass, CPT invariance transforms to M’PT conservation in either charged or neutral particles, while many other consequences are also evident.展开更多
Under the travelling wave transformation, the Camassa-Holm equation with dispersion is reduced to an integrable ordinary differential equation （ODE）, whose general solution can be obtained using the trick of one-par...Under the travelling wave transformation, the Camassa-Holm equation with dispersion is reduced to an integrable ordinary differential equation （ODE）, whose general solution can be obtained using the trick of one-parameter group. Furthermore, by using a complete discrimination system for polynomial, the classification of all single travelling wave solutions to the Camassa-Holm equation with dispersion is obtained. In particular, an affine subspace structure in the set of the solutions of the reduced ODE is obtained. More generally, an implicit linear structure in the Camassa-Holm equation with dispersion is found. According to the linear structure, we obtain the superposition of multi-solutions to Camassa-Holm equation with dispersion.展开更多
Under the travelling wave transformation, Calogero-Degasperis-Focas equation is reduced to an ordinary differential equation. Using a symmetry group of one parameter, this ODE is reduced to a second-order linear inhom...Under the travelling wave transformation, Calogero-Degasperis-Focas equation is reduced to an ordinary differential equation. Using a symmetry group of one parameter, this ODE is reduced to a second-order linear inhomogeneous ODE. Furthermore, we apply the change of the variable and complete discrimination system for polynomial to solve the corresponding integrals and obtained the classification of all single travelling wave solutions to Calogero- Degasperis-Focas equation.展开更多
Under the travelling wave transformation, some nonlinear partial differential equations such as Camassa-Holm equation, High-order KdV equation, etc., are reduced to an integrable ODE expressed by u＂ ＋p（u）（u＇）^2...Under the travelling wave transformation, some nonlinear partial differential equations such as Camassa-Holm equation, High-order KdV equation, etc., are reduced to an integrable ODE expressed by u＂ ＋p（u）（u＇）^2 ＋ q（u） = 0 whose generai solution can be given. Furthermore, combining complete discrimination system for polynomiai, the classifications of all single travelling wave solutions to these equations are obtained. The equation u＂＋p（u）（u＇）^2＋q（u） = 0 includes the equation （u＇）^2 = f（u） as a special case, so the proposed method can be also applied to a large number of nonlinear equations. These complete results cannot be obtained by any indirect method.展开更多
文摘The travelling wave group is a stable wave packet. Many surprising results are derived from it. The group is easily quantized for photons and applied, as a solution to the relativistic Klein-Gordon equation, to free particles. Further solutions to the resulting algebraic equation provide a stable wave function for free antiparticles. Consistency with the superstructure of quantum electrodynamics is obtained by an assignment to the electron antiparticle of negative mass and negative charge. Then in 5-dimensional space-time-mass, CPT invariance transforms to M’PT conservation in either charged or neutral particles, while many other consequences are also evident.
文摘Under the travelling wave transformation, the Camassa-Holm equation with dispersion is reduced to an integrable ordinary differential equation （ODE）, whose general solution can be obtained using the trick of one-parameter group. Furthermore, by using a complete discrimination system for polynomial, the classification of all single travelling wave solutions to the Camassa-Holm equation with dispersion is obtained. In particular, an affine subspace structure in the set of the solutions of the reduced ODE is obtained. More generally, an implicit linear structure in the Camassa-Holm equation with dispersion is found. According to the linear structure, we obtain the superposition of multi-solutions to Camassa-Holm equation with dispersion.
基金The project supported by Scientific Research and of Education Department of Heilongjiang Province of China under Grant No. 11511008
文摘Under the travelling wave transformation, Calogero-Degasperis-Focas equation is reduced to an ordinary differential equation. Using a symmetry group of one parameter, this ODE is reduced to a second-order linear inhomogeneous ODE. Furthermore, we apply the change of the variable and complete discrimination system for polynomial to solve the corresponding integrals and obtained the classification of all single travelling wave solutions to Calogero- Degasperis-Focas equation.
文摘Under the travelling wave transformation, some nonlinear partial differential equations such as Camassa-Holm equation, High-order KdV equation, etc., are reduced to an integrable ODE expressed by u＂ ＋p（u）（u＇）^2 ＋ q（u） = 0 whose generai solution can be given. Furthermore, combining complete discrimination system for polynomiai, the classifications of all single travelling wave solutions to these equations are obtained. The equation u＂＋p（u）（u＇）^2＋q（u） = 0 includes the equation （u＇）^2 = f（u） as a special case, so the proposed method can be also applied to a large number of nonlinear equations. These complete results cannot be obtained by any indirect method.
