Abstract: | The paper deals with a problem of developing an inverse-scattering based formalism for solving problems for the cubic nonlinear (or the modified Korteweg–de Vries (KdV)) equations: q
t
+q
xxx
+6q
2
q
x
=0, 0x<, –<t<,q
t
+q
xxx
–6q
2
q
x
=0, with the given initial and boundary conditions: q(x,0)=q(x),q(0,t)=p(t), p(t)L
1(–,). The relation between the solution of the initial-boundary value problem (1), (3), (4) and that of the KdV equation on the half-line is shown. The Cauchy problem for the cubic nonlinear equation: q
t
+q
xxx
–6|q|2
q
x
=0, 0x<, –<t<, with the given initial condition (3) is considered also. Here we solve the above problems on the half-line 0x< but with –<t<. |