Soliton resonances in four dimensional Wess-Zumino-Witten model

We present two kinds of resonance soliton solutions on the ultrahyperbolic space <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi mathvariant="double-struck">U</a:mi></a:math> for the <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"><d:mi>G</d:mi><d:mo>=</d:mo><d:mi>U</d:mi><d:mo stretchy="false">(</d:mo><d:mn>2</d:mn><d:mo stretchy="false">)</d:mo></d:math> Yang equation, which is equivalent to the anti-self-dual Yang-Mills (ASDYM) equation. We reveal and illustrate the solitonic behaviors in the four-dimensional Wess-Zumino-Witten (<h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"><h:mrow><h:msub><h:mrow><h:mi>WZW</h:mi></h:mrow><h:mrow><h:mn>4</h:mn></h:mrow></h:msub></h:mrow></h:math>) model through the sigma model action densities. The Yang equation is the equation of motion of the <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"><j:mrow><j:msub><j:mrow><j:mi>WZW</j:mi></j:mrow><j:mrow><j:mn>4</j:mn></j:mrow></j:msub></j:mrow></j:math> model. In the case of <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"><l:mi mathvariant="double-struck">U</l:mi></l:math>, the <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mrow><o:msub><o:mrow><o:mi>WZW</o:mi></o:mrow><o:mrow><o:mn>4</o:mn></o:mrow></o:msub></o:mrow></o:math> model describes a string field theory action of open <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mi>N</q:mi><q:mo>=</q:mo><q:mn>2</q:mn></q:math> string theories. Hence, our solutions on <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:mi mathvariant="double-struck">U</s:mi></s:math> suggest the existence of the corresponding classical objects in the <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"><v:mi>N</v:mi><v:mo>=</v:mo><v:mn>2</v:mn></v:math> string theories. Our solutions include multiple-pole solutions and V-shape soliton solutions. The V-shape solitons suggest annihilation and creation processes of two solitons and would be building blocks to classify the ASDYM solitons, like the role of Y-shape solitons in classification of the Kadomtsev-Petviashvili (line) solitons. We also clarify the relationship between the Cauchy matrix approach and the binary Darboux transformation in terms of quasideterminants. Our formalism can start with a simpler input data for the soliton solutions and hence might give a suitable framework for the classification of the ASDYM solitons.