| Summary: | In this paper, exact solutions of semilinear equations having exponential growth in the space variable <i>x</i> are found. Semilinear Schrödinger equation with logarithmic nonlinearity and third-order evolution equations arising in optics with logarithmic and power-logarithmic nonlinearities are investigated. In the parabolic case, the solution <i>u</i> is written as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mo>=</mo><mi>b</mi><msup><mi>e</mi><mrow><mo>−</mo><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup></mrow></msup></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo><</mo><mn>0</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>,</mo><mi>b</mi></mrow></semantics></math></inline-formula> being real-valued functions. We are looking for the solutions <i>u</i> of Schrödinger-type equation of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mo>=</mo><mi>b</mi><msup><mi>e</mi><mrow><mo>−</mo><mi>a</mi><mfrac><msup><mi>x</mi><mn>2</mn></msup><mn>2</mn></mfrac></mrow></msup></mrow></semantics></math></inline-formula>, respectively, for the third-order PDE, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mo>=</mo><mi>A</mi><msup><mi>e</mi><mrow><mi>i</mi><mo>Φ</mo></mrow></msup></mrow></semantics></math></inline-formula>, where the amplitude <i>b</i> and the phase function <i>a</i> are complex-valued functions, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Φ</mo></semantics></math></inline-formula> is real-valued. In our proofs, the method of the first integral is used, not Hirota’s approach or the method of simplest equation.
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