jQuery post to the same PHP page

I am creating a jQuery search bar, that will asynchronously send the inputted value from a form from the client side to the same page on the server side (in PHP). Then I can search my database with the value. Unfortunately when I send this value, using $.ajax, and attempt to echo out the value, I don’t get the value at all. How can I receive my value? I’ve tried print_r, var_dump, echo but to no avail.

Here is my form:

<form method = 'POST'>
  <input id="myInput" type ="text"  name ="value" 
  placeholder="press enter when finished" > 
</form> 

And here is my script to make the call. I get the value in my console when I press enter (key===13), but it seems to be the furthest my variable (value) seems to go.

$(document).ready(function(){
    $('#myInput').bind('keypress', function(e){ 
        if(e.keyCode ===13){
        let value = $(this).val().toLowerCase();
        console.log(value);
            
        e.preventDefault();  //stops the damn page refreshing
            
         $.ajax({
            type: "POST",
            data: value
            
        
        });

        };
    });
});

I haven’t put the URL in the AJAX call for a reason: I’m sending to the same page.

Here is the code (further up the page) that I’m using to echo out the posted value, on the same page.

if(isset($_POST['value'])) { echo $_POST['value']; exit;  }

No matter what I seem to do, my value isn’t sent to the page! I’ve tried to add in the URL, I’ve added “success” to the call…. nothing seems to work.

How can I successfully send my data?