Stability of Solitary-Wave Solutions of Systems of Dispersive Equations

The present study is concerned with systems (Formula presented.),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.P(u,v)=Au2+Buv+Cv2,Q(u,v)=Du2+Euv+Fv2,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