Applied Mathematics and Optimization, Vol.75, No.1, 27-53, 2017
Stability of Solitary-Wave Solutions of Systems of Dispersive Equations
The present study is concerned with systems of Korteweg-de Vries type, coupled through their nonlinear terms. Here, u = u(x, t) and v = v(x, t) are real-valued functions of a real spatial variable x and a real temporal variable t. The nonlinearities P and Q are homogeneous, quadratic polynomials with real coefficients A, B,..., viz. in the dependent variables u and v. A satisfactory theory of local well-posedness is in place for such systems. Here, attention is drawn to their solitary-wave solutions. Special traveling waves termed proportional solitary waves are introduced and determined. Under the same conditions developed earlier for global well-posedness, stability criteria are obtained for these special, traveling-wave solutions.