how to upload file asynchronously on dropbox by using php sdk with the help of ajax?

If you want to upload a file using AJAX client-side in the browser, you can use the JavaScript SDK:

https://www.dropbox.com/developers/datastore/sdks/js

There's a tutorial here that shows how to get started:

https://www.dropbox.com/developers/datastore/tutorial/js

The full documentation can be found here:

https://www.dropbox.com/developers/datastore/docs/js

For example, here's the method for uploading a file:

https://www.dropbox.com/developers/datastore/docs/js#Dropbox.Client.writeFile

Alternatively, if you do want to have the API call occur on your server using the PHP SDK, you could potentially upload the file from the browser to your server using an AJAX call and then pass the data to the uploadFile method of the PHP SDK:

http://dropbox.github.io/dropbox-sdk-php/api-docs/v1.1.x/class-Dropbox.Client.html#_uploadFile

Since you'd be using an AJAX call to pass the data to your server, you wouldn't need a full page reload. This part isn't specific to the Dropbox API though.