The Solutions Of The Generalized B-family Equation

Nonlinear Science, which has solution theory, fractal and chaos as its main parts, is the subject of studying the common futures of nonlinearity. Nonlinearity is universal and important. Most nonlinear problems can be described by nonlinear equations, which generally include nonlinear.The paper introduces the family of evolutionary 1+1 PDEs that describe the balance between convection and stretching in the dynamics of 1D nonlinear waves in fluids, here u = g*m. This convolution (or filtering)relates velocity u to momentum density m by integration against the kernel g(x).We shall let g(x) be an even function, so that u and m have the same parity under spatial reflection. This equation is both reversible in time and parity invariant. We shall study the effects of the balance parameter b and the kernel g(x) on the solitary wave structures, and investigate their interactions for v = 0 and for small viscosity v≠0.Whenf b = 0, peakons, ramps and cliffs are discussed in the paper. Moreover, whenf b≠0, general solution is given. Especially, with b = 3 and b = -1, the paper obtains the exact solutions of the generalized b-family equation.