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I can't find 4kW single phase to single phase drive in the market. Is thre any way to run a single phase motor with VFD which has 3-phase output?

Square D has just such a drive:

ALTIVAR 11 AC Drives

The link is:
http://ecatalog.squared.com/techlib/docdetail.cfm?oid=090089268 0136d83

A clarification on the Altivar Drive (and most other brands & models). AC Drives are designed to only control three-phase motors. Some smaller models VFD such as the Altivar 11 will accept single-phase input and convert the output to three-phase power. I am unaware of any manufacture that still produces AC Drives for single-phase motors.

As Alan mentioned earlier, single-phase motors have a variety of characteristics (permanent split capacitor, capacitor start/run…) that makes them difficulty to control from a VFD. The motor needs to be changed to 3-phase if it is to be used with a AC Drive. A common application is a 1Ph 120Vac input where the Drive produces 3Ph 230V output. The Altivar 11 series as well as others are capable of this.

There are companies still making drives for single phase motors (lots of single-phase fan motors out there to be controlled). Dwyer makes its "FC-1000", and Control Resources Inc makes their "Stratus" drives just for this purpose. For my part, I am moving to the CRI Stratus to replace the FC-1000. The Straus can be used to drive a single phase motor (single phase Input) or a 3-phase motor (single or 3-phase input), so I'll be stocking ONE drive instead of several. Info here: http://www.controlres.com/stratus.htm

There are too many problems trying to use variable frequency drives with single phase motors - internal starting coils and switches, etc. Replace the single phase motor with a three phase motor.

You can, however, run many small three-phase drives from single phase power.

how can I run a 3 phase vfd on single phase?

Three phase drives will run off 1 phase power because they convert 3 phase to DC before converting it back to 3 phase variable frequency.

Running on single phase just reduces the power available to the DC section so you have to de-rate the drive accordingly.

It's also possible to get drives designed especially for single phase input, some that boost the voltage e.g. from 120 to 208

Roy

You can use Invertek drives. They are based in the UK. Decide between the Optidrive E1 or the Optidrive E2.