Prohibitions caused by nonlocality for nonlocal Boussinesq‐KdV type systems

It is found that two different celebrate models, the Korteweg de‐Vrise (KdV) equation and the Boussinesq equation, are linked to a same model equation but with different nonlocalities. The nonlocal KdV equation can be derived in two ways, via the so‐called consistent correlated bang companied by the parity and time reversal from the local KdV equation and via the parity and time reversal symmetry reduction from a coupled local KdV system which is a two‐layer fluid model. The same model can be called as the nonlocal Boussinesq system if the nonlocality is changed as only one of parity and time reversal. The nonlocal Boussinesq equation can be derived via the parity or time reversal symmetry reduction from the local Boussinesq equation. For the nonlocal Boussinesq equation, with help of the bilinear approach and recasting the multisoliton solutions of the usual Boussinesq equation to an equivalent novel form, the multisoliton solutions with even numbers and the head on interactions are obtained. However, the multisoliton solutions with odd numbers and the multisoliton solutions with even numbers but with pursuant interactions are prohibited. For the nonlocal KdV equation, the multisoliton solutions exhibit many more structures because an arbitrary odd function of x+t can be introduced as background waves of the usual KdV equation.