A rather contrived example of an issue I'm running into trying to use one string to search over first and last names together

create_test_index.sh

#!/bin/bash

echo 'Deleting index if it exists'
curl -XDELETE 'http://127.0.0.1:9200/test'
echo
echo 'Creating test index'
curl -XPUT 'http://127.0.0.1:9200/test/?pretty=1'  -d '
{
   "mappings" : {
      "user" : {
         "properties" : {
            "last_name" : {
               "fields" : {
                  "last_name" : {
                     "type" : "string",
                     "analyzer" : "full_name"
                  }
               },
               "type" : "multi_field"
            },
            "first_name" : {
               "fields" : {
                  "first_name" : {
                     "type" : "string",
                     "analyzer" : "full_name"
                  }
               },
               "type" : "multi_field"
            }
         }
      }
   },
   "settings" : {
      "analysis" : {
         "analyzer" : {
            "full_name" : {
               "filter" : [
                  "standard",
                  "lowercase",
                  "asciifolding"
               ],
               "type" : "custom",
               "tokenizer" : "standard"
            }
         }
      }
   }
}'
echo
echo 'Populating test index'
curl -XPOST 'http://127.0.0.1:9200/_bulk?pretty=1'  -d '
{"index" : {"_index" : "test", "_type" : "user"}}
{"last_name" : "John", "first_name" : "John"}
{"index" : {"_index" : "test", "_type" : "user"}}
{"last_name" : "Smith", "first_name" : "John"}
{"index" : {"_index" : "test", "_type" : "user"}}
{"last_name" : "Sánchez", "first_name" : "John"}
{"index" : {"_index" : "test", "_type" : "user"}}
{"last_name" : "Smitt", "first_name" : "Joe John"}
{"index" : {"_index" : "test", "_type" : "user"}}
{"last_name" : "John", "first_name" : "Smith"}
'
echo
echo 'Done'

query_test_index.sh

#!/bin/bash
echo
echo "User searches for 'John Smith' in search bar"
echo "Expected result 'John Smith' first"
echo "Actual result:"
curl -XGET 'http://127.0.0.1:9200/test/user/_search?pretty=1'  -d '
{
    "query" : {
        "bool" : {
            "should" : [
                {
                    "text" : {
                        "first_name" : "John Smith"
                    }
                },
                {
                    "text" : {
                        "last_name" : "John Smith"
                    }
                }
            ]
        }
    }
}
' 
echo
echo
echo "User searches for 'John Sánchez' in search bar"
echo "Expected result 'John Sánchez' first"
echo "Actual result:"
curl -XGET 'http://127.0.0.1:9200/test/user/_search?pretty=1'  -d '
{
    "query" : {
        "bool" : {
            "should" : [
                {
                    "text" : {
                        "first_name" : "John Sánchez"
                    }
                },
                {
                    "text" : {
                        "last_name" : "John Sánchez"
                    }
                }
            ]
        }
    }
}
'
echo 
echo
echo "User searches for 'Smith John' in search bar"
echo "Expected result 'Smith John' first"
echo "Actual result:"
curl -XGET 'http://127.0.0.1:9200/test/user/_search?pretty=1'  -d '
{
    "query" : {
        "bool" : {
            "should" : [
                {
                    "text" : {
                        "first_name" : "Smith John"
                    }
                },
                {
                    "text" : {
                        "last_name" : "Smith John"
                    }
                }
            ]
        }
    }
}
' 
echo