We used to have this problem too. This is one of those situations where using weedkiller is the best option.

Our local council has extensive advice on controlling Japanese Knotweed. Maybe yours has too. This is what our council advises -
http://www.cornwall.gov.uk/default.aspx?page=13805

Your neighbour is responsible for controlling the knotweed on their property. If they're not doing this you could ask them to - if they don't you may need to involve your local council.
This page -
http://www.cornwall.gov.uk/default.aspx?page=19756
says -
"Allowing Japanese knotweed to spread onto neighbouring land could be considered either a statutory or private nuisance."
It is possible to take legal action against them if they refuse to deal with it, but it would be best to approach them about it first.

There is a very useful online version of the leaflet "Guidance for identification and control" if you click on the link on this page.
http://www.cornwall.gov.uk/default.aspx?page=19740

Japanese Knotweed is considered the most invasive plant in Britain. It grows up to 10cm per day, reaching 3m in height. But it is below ground where this invader causes the biggest problems, extending to 3m in depth and 7m in all directions. This makes it a serious threat to construction where it can have devastating consequences damaging foundations, drains and other underground services. JK has a large underground network of roots (rhizomes). To eradicate the plant the roots need to be killed. Picking the right herbicide is essential, as it must travel through the plant and into the root system below.Firstly make sure that this is JK? then seek expert advice as its costly and you need to sort out who pays?

Many thanks I will take a photo next time I am there.

As we volunteer in this community garden we need to be careful with plants.

I know there is a super-strength contact weedkiller available commercially because our Council used it to eradicate Japanese knotweed (it only took two applications). Unfortunately it is not available to amateur gardeners but you might try to persuade the Council to intervene as it is threatening a Community Garden

If you have a lot of time and patience and not too much knotweed its possible to cut down each individual stem and pour deep root weedkiller directly into it. My son did this fairly successfully, but if it is invading from outside the garden the only real option is to get it all eradicated. It needs constant vigilance to discourage it though.