Exact special solutions with solitary patterns for the nonlinear dispersive K(m,n) equations
Abstract In this paper, we study the genuinely nonlinear dispersive K(m,n) equation, ut−(um)x+(un)xxx=0, which exhibits solutions with solitary patterns. Exact solutions that create solitary patterns having cusps or infinite slopes are developed. The nonlinear equation K(m,n) is addressed for two different cases, namely when m=n=odd integer and when m=n=even integer. General formulas for the solutions of these cases of the K(m,n) equations are established.
Abstract This paper follows the line of research aimed to develop randomized algorithms for probabilistic analysis and design of control systems. In particular, a result for the generation of real matrix samples uniformly distributed in the spectral norm ball is presented. To this end, the distribution of the singular values of random uniform matrices is first studied. The sample generation is then based on the conditional densities method. A key result of this paper is the computation in closed form of the marginal densities of the singular values, leading to a polynomial-time method for the real matrix sample generation problem.